Frobenius-Artin Algebras and Infinite Linear Codes

FROBENIUS-ARTIN ALGEBRAS AND INFINITE LINEAR CODES

MIODRAG CRISTIAN IOVANOV

Abstract. We generalize the results on finite Frobenius rings of T. Honold [Arch. Math (Basel) 76, no. 6 (2001), 406–415] and some classical results of Nakayama [Ann. Math, 1939 & 1941] on Frobenius algebras over fields, and the results of J.A. Wood [Proc. Amer. Math. Soc. 136, no. 2 (2008), 699–706] & [Amer. J. Math 121, no.3 (1999), 555–575] on linear codes and finite Frobenius rings, to the setting of Artin algebras, and provide a unifying context for these results. We show that an Artin algebra is Frobenius if and only if its the socle and top are isomorphic only as left modules (equivalently, as right modules). We show that an Artin algebra A satisfies the MacWilliams code equivalence property if and only if A is a product of a finite Frobenius ring and a quasi-Frobenius ring with no nontrivial finite representations.

Introduction Frobenius algebras have emerged with the following natural question asked by G. Frobe- nius. Given a finite dimensional K-algebra A, consider B = {v1, . . . , vn} a basis of A. Let η : A → Mn(K), respectively, ρ : A → Mn(K) be the representations of A given by η(a) = the matrix (with respect to the basis B) of the left, respectively, right, multiplication by a. Frobenius’ question is to characterize algebras for which the representations η and ρ are equivalent (see [CR]). This is equivalent to asking that the algebra A is isomorphic as a left ∗ (and equivalently, also as right) A-module with its K-dual A . The class of Frobenius algebras is quite extensive. The first important examples were group algebras of finite groups: while in characteristic zero a group algebra is semisimple, this is not true if char(K) divides |G|, but the group algebra is always Frobenius, which is important in modular representation theory. Frobenius algebras have appeared from many fields, such as topology (the cohomology ring of a compact oriented manifold with coefficients in a field is a Frobenius algebra by Poincaré duality), topological quantum field theory (there is a one-to-one correspondence between 2-dimensional quantum field theories and commutative Frobenius algebras; see [Ab’96]), ge- ometry (Frobenius manifolds), Hopf algebras (a finite dimensional Hopf algebra is a Frobenius algebra), representation theory (and tensor categories), cryptography and codes, and have thus constituted the subject of much research. Hopf and quasi-Hopf algebras also produce many interesting examples, and there may even be infinitely many isomorphism types of Hopf algebras of a certain dimension [BDG99]. Besides the above mentioned examples, there are large classes of examples which can be obtained from combinatorial objects such as quivers and PO-sets; see [I’12]. Frobenius algebras have the important homological property that injective and projective modules coincide. Finite dimensional algebras, or more generally, rings for which a module is

Key words and phrases. Frobenius algebra, Artin algebra, linear codes, quasi-Frobenius, finite ring. 2010 Mathematics Subject Classification. 16A39, 22B05, 28E75, 43A25, 94B05. 1 2 MIODRAG CRISTIAN IOVANOV projective if and only if is injective are called quasi-Frobenius. By extension, categories with this property are called Frobenius categories. Thus, quasi-Frobenius algebras are the cate- gorical generalization of Frobenius algebras. Frobenius rings were introduced as an abstract counterpart of Frobenius algebras, in the absence of a basefield K. A ring R is Frobenius if it is quasi-Frobenius and the (left or right) socle of R is isomorphic to R/Jac(R) on the left and on the right. Hence, the Frobenius property for rings is equivalent to the categor- ical quasi-Frobenius plus some multiplicity condition for simples (or, equivalently, for the indecomposable direct summands of the ring). Much of this theory was developed by the landmark papers of Nakayama [N’39, N’41] and later in [N’43, N’49]. On the other hand, finite Frobenius rings have recently proved to be of interest for cryptography. Although the algebraic theory of error correcting codes is originally developed over F2, the study of linear codes over finite rings has become increasingly more important (see, for example, [CHKSS’93]). Moreover, it appears that codes over finite Frobenius rings have many of the features present for linear codes over F2. At the same time, finite Frobenius rings admit characterizations that are similar to those of Frobenius algebras; for example, a finite ring is Frobenius if its left socle and its left top are isomorphic (equivalently, the right socle and top are isomorphic), by a result of Honold [Ho’01]. J.A. Wood proved that over a finite Frobenius ring, linear codes have the MacWilliams extension property [M] for linear codes in [W’99] (see also [WW’96]; [GS’00] later also gives a more combinatorial proof). Wood’s proof relies on character theory, but also on some fine key ring theoretic observations. A partial converse (for commutative rings) was also proved in [W’99], and other partial converses have been proved in [DL-P1’04, DL-P2’04]. As a culmination of these results, the complete converse statement, that a finite ring that satisfies MacWilliams’ equivalence theorem (or extension property) for linear codes is necessarily Frobenius, is proved in [W’08]. The purpose of this note is to give generalizations of the above results to arbitrary rings, and also provide a unifying context for several results on Frobenius rings and Frobenius algebras which work seemingly independent at the same time for finite rings and finite dimensional algebras. For example, the result of [Ho’01] is valid for finite dimensional algebras by the work of Nakayama [N’39, N’41, N’43, N’49]. The natural context generalizing the finite rings realm and finite dimensional algebras at the same time is that of Artin algebras. Recall that an Artin algebra is an algebra A over a commutative artinian ring K, such that A is finitely generated over K. We define Frobenius algebras over such rings K, via the self- ∨ duality functor (−) of K-mod, in a way that parallels Frobenius algebras over fields: A is Frobenius if A =∼ A∨ as left A-modules. In Section 1, we show that such an algebra is an Artin algebra, and that A is Frobenius if and only if it is a Frobenius ring. In our first main result, Theorem 1.7, we give a short proof to show that A is a Frobenius-Artin algebra if and only if the left socle and left top of A are isomorphic (equivalently, right socle and right top are isomorphic), generalizing the above mentioned main result of [Ho’01] and the corresponding result of Nakayama on algebras over fields. Next, we turn to an infinite version of MacWilliams’ code equivalence theorem (or extension property for linear codes). We generalize the main result of [W’99] to arbitrary rings, and show that in the infinite situation the quasi-Frobenius property is the one more closely related to the extension property for linear codes. In Section 4 we show that a ring which is a product of a finite Frobenius ring and a quasi-Frobenius ring with no non-trivial modules has the extension property for (left and right) linear codes (Theorem 4.2 and Corollary 4.6). We FROBENIUS-ARTIN ALGEBRAS AND INFINITE LINEAR CODES 3 give two proofs for this, one that uses abstract harmonic analysis on compact groups (Haar measure, Pontryagin duality) and is the generalization of the method proposed by Wood, and a second one, which is of a more combinatorial flavor, and is based on some results regarding modules which are finite unions of submodules. We also prove a converse for Artin algebras in Section 3, converse which can be regarded as an extension of the result of [W’08]: an Artin algebra A which has the extension property for left (or right) linear codes decomposes as ∼ A = Af × A∞, with Af finite Frobenius and A∞ a quasi-Frobenius (Artin) algebra with no finite nontrivial modules (Theorem 3.7). We show that in fact only the extension property for semisimple modules is enough to imply the conclusion of this converse. We also note that a left artinian ring which has the left and right extension property for simple modules is quasi-Frobenius. On the way, we find, as direct consequences, some other known results on QF rings. Also, in section 2, we prove a decomposition theorem for Artin algebras and QF rings, into product of rings having all simple modules either of the same cardinality or finite. Our main results can then be summarized by the following

Theorem 0.1. ∼ (I) Let A be an Artin algebra. Then A is Frobenius if and only if soc(AA) = A/Jac(A) as ∼ left modules, equivalently, soc(AA) = A/Jac(A) as right modules. (II) If A = Af × A∞ with Af a ﬁnite Frobenius ring and A∞ a quasi-Frobenius ring with no nontrivial ﬁnite modules, then A has the extension property for left and right linear codes (i.e. the MacWilliams equivalence of linear codes) . (III) Conversely, if an Artin algebra A has the left extension property for linear codes, then it decomposes as A = Af × A∞ as in (II) above.

With a general audience in mind, we recall most of the deﬁnitions and terminology used, as well as techniques and results required from ring theory, Frobenius algebras, measure theory and the theory of compact groups.

1. Frobenius Algebras

Throughout this paper, unless otherwise specified, K will be a commutative artinian ring. Let E be a minimal injective cogenerator of K-Mod (the sum of the injective hulls of the simple ∨ K-modules). By Matlis duality, the functor (−) = HomK(−,E) from the category of finitely generated (equivalently, artinian, noetherian, of finite length) K-modules to itself is a duality of categories. We fix some notation. For a module M over a ring A, we denote by soc(M) the socle of M, the largest semisimple submodule of M, and dually, top(M) = M/Jac(M), where Jac(M) denotes the Jacobson radical of M. We will write J = Jac(A) for short for the ∗ Jacobson radical of the ring A. For a left A-module M, we also denote by M = HomA(M,A) ∗ its A-dual right A-module. For a right module N, let us denote N = HomA(N,A) its A-dual, which is a left A-module. For a K-module M we write lK(M) for the length of M. We recall the following results which give equivalent definitions of Frobenius and Quasi-Frobenius (QF) rings:

Theorem 1.1 (Quasi-Frobenius rings). Let R be a ring. The following are equivalent: (i) R is left noetherian and left selﬁnjective. (ii) R is left artinian and left selﬁnjective. (iii) R is left artinian and left cogenerator. 4 MIODRAG CRISTIAN IOVANOV

(iv) right versions of (i)-(iii). A ring satisfying the above equivalent conditions is called quasi-Frobenius ring, or QF ring for short. For a ring R, let S denote a set of representatives for the types of isomorphisms of simple left R-modules. Given a QF ring R, there is a permutation σ of the set S, defined as follows. For each S ∈ S, let P (S) be its projective cover. Since P (S) is indecomposable, and also injective (since R is selfinjective), it has simple socle, isomorphic to some S0 ∈ S. One defines σ(S) = S0, so σ(S) =∼ soc(P (S)). This σ is called the Nakayama permutation (note: sometimes a slightly different but equivalent version of this permutation is considered). For any semilocal ring R, for each S ∈ S, let us denote by nS the multiplicity of S in R/J; also, denote mT the multiplicity of a simple right R-module T in R/J. We will also denote, for a ring R with finite socle, by kS the multiplicity of S in socR(R). The following is [L, 16.14], and characterizes a stronger class of rings. Theorem 1.2 (Frobenius rings). Let R be an artinian ring, and J = Jac(R). The following are equivalent: ∼ ∼ (i) R is QF and soc(RR) = top(RR) (i.e. soc(RR) = R/JR). ∼ ∼ (ii) R is QF and soc(RR) = top(RR) (i.e. soc(RR) = RR/J). (iii) R is QF and nS = nσ(S) for each S ∈ S. ∼ ∼ (iv) soc(RR) = top(RR) and soc(RR) = top(RR). If R satisfies these equivalent conditions, then R is said to be a Frobenius ring. We also recall the following well known

Definition 1.3. Let K be a field. In the case when K is a field, if A is K-algebra, then ∼ ∨ A is said to be Frobenius if and only if A = A = Hom(A, K) as left (equivalently, right) A-modules. In this case of the definition above, A∨ coincides to the vector space dual of A. It is known that for an algebra, A is a Frobenius algebra if and only if A is a Frobenius ring. The main aim of this section is to generalize several results of [Ho’01, N’43, W’99], and provide a unified context for results on Frobenius algebras over fields, and finite Frobenius rings. Recall that K is a fixed commutative artinian ring. We introduce the following ∼ ∨ Definition 1.4. Let A be a K-algebra. We say A is a Frobenius algebra if and only if A = A as left A-modules. ∼ ∨ We first recall a well known fact: if V is K-module, and V = V , then V is finitely generated. To see this, one can reduce to the local case since K is a product of local commutative artinian rings. So, assume m is maximal in K and L = K/m. If V is not finitely generated, (α) L let V0 = soc(V ) have (infinite) length α (i.e. dimension α over L), so V0 = V = L. α Then V =∼ V ∨ implies V ∨ =∼ (V ∨)∨, so V =∼ (V ∨)∨. Since L(α) ⊂ V , there is an epimorphism V ∨ → (L(α))∨ = Lα = Q L. This is a semisimple module of length equal to some cardinality α α α β; we have β = dimL(L ). Note that β ≥ 2 , since it is well known that for over fields, dim(Lα) ≥ 2α (see, for example, [J, Chapter IX] for a more general statement). Furthermore, ∨ dualizing the epimorphism V ∨ → L(β) we obtain a monomorphism L(β) ,→ (V ∨)∨, which shows that the socle of (V ∨)∨ =∼ V has length at least 2β. Hence, α ≥ 2β > β ≥ 2α > α, which is not possible. Therefore α is finite. FROBENIUS-ARTIN ALGEBRAS AND INFINITE LINEAR CODES 5

Proposition 1.5. The following are equivalent for a K-algebra A over a commutative artinian ring K. (i) A =∼ A∨ as left A-modules. (ii) A =∼ A∨ as right A-modules. (iii) A is a Frobenius ring. Moreover, if the above hold, then A is an Artin algebra.

Proof. Let us first notice that (i) or (ii) imply that A is a QF ring. First, by the above note, A is an Artin algebra. Now, since (−)∨ is a duality on finitely generated A-modules, A∨ is an injective left A-module, and so A is selfinjective. Hence, A is QF. Now, let us show that if A is QF, then A =∼ A∨ as left modules if and only if A is a Frobenius ring. This completes (i)⇔(iii), and (ii)⇔(iii) is similar.

For right simple modules T denote mT the multiplicity of T in A/J. Using the selfduality of A/J, it is standard to note that for a simple left A-module S, nS = mS∨ . Now, for each ∨ simple S ∈ S, the multiplicity of P = P (S) in A is nS. Let Q = P (S) . Since σ(S) = soc(P ), σ(S)∨ = top(P ∨) = top(Q). The multiplicity of P = Q∨ in A∨ is the multiplicity of Q in ∼ ∨ A, i.e. mσ(S)∨ . This equals nσ(S). This shows that A = A as left modules if and only if the multiplicities of all P ’s agree, i.e. if and only if nS = nσ(S) for all S ∈ S. By the characterization Theorem 1.2, this ends the proof. The above theorem entitles us to call an Artin algebra which is a Frobenius ring a Frobenius algebra, or Frobenius-Artin algebra. We note another characterization that parallels one from Frobenius algebras over ﬁelds. We need the following remark: if A is an Artin algebra, P is an indecomposable projective with top(P ) = S, soc(P ) = L then EndA(P ) is a local ring with maximal ideal M = {f | f(L) = 0} = {f | f(P ) ⊆ JM}. The quotient EndA(P )/M is ∼ therefore easily seen to be isomorphic to both EndA(S) and EndA(L), so EndA(L) = EndA(S). ∼ n ∼ Since S = EndA(S)S as K-modules and EndA(S) = EndA(σ(S)), we see that nS = nσ(S) if and only if lK(S) = lK(σ(S)). Hence, we have Theorem 1.6. Let A be an Artin K-algebra which is a QF-ring. Then A is a Frobenius algebra if and only if lK(soc(P )) = lK(top(P )) for any projective indecomposable left (equivalently, right) A-module P .

We now give the main result of this section. ∼ ∼ Theorem 1.7. Let A be an Artin algebra. If soc(AA) = top(AA), then soc(AA) = top(AA) and A is a Frobenius algebra. ∼ Proof. First, let us note that the isomorphism soc(AA) = top(AA) implies that each projective indecomposable left A-module P has simple socle. We may then introduce the function (a priori, not necessarily a permutation) σ : S → S, by σ(S) = soc(P (S)) (P (S) is the pro- ∼ jective cover of S). The isomorphism soc(AA) = top(AA) also shows that all S ∈ S appear in soc(AA), which is a direct sum of modules of the type σ(S). Hence, σ is onto, and therefore a bijection.

As before, let kS be the multiplicity of S ∈ S in soc(AA) and mT be the multiplicity of the right module T in A/J. The hypothesis shows that kL = nL for L ∈ S. Obviously, ∼ ∨ nS = kσ(S), so nS = kσ(S) = nσ(S) = mσ(S)∨ . The last equality follows since A/J = (A/J) , ∨ ∨ as A/J is a semisimple Artin algebra. Let QS be the projective cover of σ(S) . Then QS is 6 MIODRAG CRISTIAN IOVANOV

∨ injective with socle σ(S), so P (S) embeds in QS, since σ(S) is essential in P (S). Therefore, we have an embedding M nS M ∨ nS A = P (S) ,→ (QS) S∈S S∈S L But since AA = [mσ(S)∨ ] · QS, and mσ(S)∨ = nS, the monomorphism above gives an S∈S embedding of left A-modules A,→ A∨ ∨ ∼ ∨ But since lK(A) = lK(A ), we have obtained an isomorphism A = A . Hence, A is Frobenius ∼ and soc(AA) = top(AA). Corollary 1.8. If A is a finite ring or a finite dimensional algebra over a field L, such that ∼ soc(AA) = top(AA), then A is a Frobenius ring (algebra). Proof. If A is a finite ring, then it has finite characteristic n, and it can be considered as an Artin algebra over K = Z/n. In the second case, take K = L. The above statement for Frobenius algebras over fields is proved in Nakayama’s papers [N’39, N’41, N’43]. The statement for finite rings is proved in [Ho’01]. Note however that it does not hold for arbitrary rings, as Nakayama [N’43] provided an example of an artinian ring with ∼ soc(AA) = top(AA), but soc(AA) =6 socAA. We will see in what follows that this property is closely related to the extension property for linear codes.

2. Artin algebras: set theoretic considerations We note an interesting fact about artinian rings: namely, they decompose into products of rings for which the simple modules have the same cardinality. For a ring A denote J = Jac(A) its Jacobson radical, and let again S be a set of representatives for the types of isomorphism simple left A-modules. For two cardinalities α, β, we will write α β if either they are both ﬁnite, or α ≤ β as cardinals. We write α ≈ β if either they are both ﬁnite, or α = β as cardinals (cardinal numbers). For a set X denote by |X| its cardinality.

Proposition 2.1. Let A be a left artinan ring. 1 (i) If S, T are simple left A-modules such that Ext A(T,S) 6= 0, then |T | |S|. (ii) Let A be a left artinian ring, and assume that the injective hulls of simple modules have ﬁnite length (i.e. A-Mod has a ﬁnitely generated injective cogenerator). If S, T are simple left A-modules such that Ext 1(T,S) 6= 0, then |T | ≈ |S|.

Proof. (i) Let P be the projective cover of T . There is a non-split sequence of A-modules 0 → S → V → T → 0, and an epimorphism P → T , which then lifts to an epimorphism P → V . It follows that S is a quotient of JP = Jac(P ). Let X = P/Y be the maximal quotient of P for which JP/Y is isomorphic to a power of S, say JP/Y =∼ Sk (equivalently, Y is the minimal submodule of JP such that JP/Y is a power of S). Now note that there is a natural map Γ : End(P ) → End(JP/Y ) (all Hom spaces here will be considered over the ring A). Indeed, for ψ ∈ End(P ), then ψ(JP ) ⊆ JP , and it is easy to see that there is a canonical monomorphism JP/ψ−1(Y ) ,→ JP/Y . Thus, JP/ψ−1(Y ) is a power of S, so Y ⊆ ψ−1(Y ). This shows that ψ induces a morphism ψ : JP/Y → JP/Y , and we put Γ(ψ) = ψ. It is standard to see that there is also a natural map Π : End(X) → End(T ), given by FROBENIUS-ARTIN ALGEBRAS AND INFINITE LINEAR CODES 7 reduction modulo JP . Moreover, Π is surjective (since P is projective), and End(P ) is a local ring with maximal ideal ker(Π), since P is a projective cover of T . Both Γ and Π are nonzero morphisms of rings; moreover, ker(Γ) ⊆ ker(Π), by the maximality of ker(Π), so there is a homomorphism End(T ) → End(Sk), which is injective (since End(T ) is a skewfield). Since for a simple A-module L over an artinian ring A, the cardinality of End(L) is either the same as that of L, or they are both finite, it follows that |T | |S|. (ii) One can use the same argument as above dually with injective hulls (which are of finite length), and show that if Ext 1(T,S) 6= 0, then we also have |S| |T |. Combined with (i), this yields |T | ≈ |S|. Proposition 2.2. Let A be a ring. Assume that A is either an Artin algebra, or a QF ring. 1 If S, T are simple A-modules such that Ext A(T,S) 6= 0, then |S| ≈ |T |, where |X| denotes the cardinality of the set X. Proof. The condition on injective hulls of Proposition 2.1(ii) is satisfied in both situations, i.e. when A is an Artin algebra and when A is a QF ring. Corollary 2.3. If A is an Artin algebra or a QF ring, then A is a finite direct product of Q rings A = Ai, such that each Ai has the property that either its simple modules have equal i infinite cardinalities, or they are all finite (i.e. Ai is finite in that case). Proof. Partitioning the set of left simples into classes containing simples of the same cardinality, and one class containing finite simples, by the previous proposition we have that Ext (T,S) = 0 for simples S, T in different classes. Using block theory (block decomposition), the conclusion follows immediately. 3. Linear codes and Frobenius-Artin algebras We provide here the connection between the (quasi-)Frobenius property of Artin algebras and linear codes. For the convenience of the reader, we briefly recall a few basic facts on linear codes, for which we refer to the textbooks [B, H]. We introduce these notions in the greater generality of rings as opposed to binary codes over F2, as in [W’99]. A left (right) linear code over a ring A is a left (right) A-submodule L of An, for some positive integer n. n For an element x = (x1, . . . , xn) ∈ A , the Hamming weight of x is, by definition, the number of nonzero components of x: wt(x) = |{i|xi 6= 0}|. A left (right) isometry of An is an A-linear automorphism T of An which preserves Hamming weight, i.e. wt(T (x)) = wt(x), for all x ∈ An. A left monomial transformation T of An is an n T ∈ Aut A(A ) of the form

T (x1, . . . , xn) = (xσ(1)u1, . . . , xσ(n)un) for some permutation σ of {1, 2, . . . , n}, and units u1, . . . , un of A. Two linear codes L1,L2 n are said to be equivalent if there exists an isometry T of A such that T (L1) = L2. We note the following analogue of [W’99, Proposition 6.1] is true for any ring A:

n Proposition 3.1. A left A-module endomorphism T ∈ EndA(A ) is an isometry if and only if it is a left monomial transformation.

Obviously, if two linear codes L1,L2 are equivalent via the isometry T , then the A-module isomorphism T : L1 → L2 is an isomorphism. The converse has also been considered in 8 MIODRAG CRISTIAN IOVANOV literature, and has been called the extension property. It concerns the situation when if two linear codes that are isomorphic through a weight preserving isomorphism, must in fact be equivalent (i.e. are isomorphic through a monomial transformation).

Deﬁnition 3.2. We say that a ring A has the left extension property or, following [DL-P1’04, DL-P2’04], that it is a left MacWilliams ring (in honor of F.J.MacWilliams) if for every left linear code L ⊆ An, and an A-module monomorphism f : L → An, there is a left isometry of An extending f.

We note that the above condition is equivalent to asking that whenever L1,L2 are two linear n n n codes in A , and f : L1 → L2 is an isomorphism, then there is an isometry F : A → A extending f. We first aim to give a short proof of the fact that rings with the extension property must be quasi-Frobenius. For finite rings, it is shown in [W’08] that rings with the extension property must be Frobenius. For convenience and to fix notation, we briefly recall some Wedderburn-Artin theory. If B is a semisimple ring, with simple left modules S1,...,Sn, then B = B1 × · · · × Bn, and the Q annihilator of Sk is Ik = Bi. Moreover, Bi = Bei, with ei central idempotents. We will i6=k use the following fact: if e, f are primitive idempotents of B such that Be =6∼ Bf, then ef = 0. Indeed, Be, Bf are nonisomorphic simple modules, so they have different annihilators. Thus, there are i, j, i 6= j such that e = eei = eie and f = fej = ejf. Hence, ef = eeiejf = 0. Note that a ring A has the left extension property for simple modules (simple linear codes) if and only if every morphism from a simple left ideal I of A is given by right multiplication by an invertible element of A. Rings for which every morphism ϕ : I → A from a simple ideal of A is given by right multiplication were studied before, and called left mininjective rings [Ha’82, Ha’83, NY’97]. Obviously, if a ring has the left extension property for simple modules, then it is left mininjective. It is shown in [NY’97, Corollary 4.8] that a left and ring mininjective ring which is left Artinian, is QF. Thus, it follows that a ring that has both the left and right extension properties for linear codes and is left Artinian is necessarily QF. We will show that the left extension property is enough though; for this, we need a few easy propositions, which, in particular, will also yield an independent short proof of the above mentioned result of [NY’97].

Proposition 3.3. Let A be a left artinian ring which has the left extension property for simple modules. Then for each simple left A-module S, there is, up to isomorphism, at most one indecomposable projective P such that Hom(S, P ) 6= 0.

Proof. Assume otherwise, that f : S → P and f 0 : S → P 0 are embeddings of the simple S into indecomposable projectives P,P 0 with P =6∼ P 0. We may obviously assume P and P 0 are direct summands of A. Using the extension property for f, f 0, we ﬁnd a ∈ U(A) such that f(x) = f 0(x)a. Then Q = P 0a is also an indecomposable projective (since a is invertible), and we have P ∩Q ⊇ f(S) 6= 0 so P ∩Q 6= 0. Let P = Ae and Q = Af with primitive idempotents e, f. Since P =6∼ Q are projective indecomposable, we have top(P ) =6∼ top(Q). Denote x ∈ A/J the image of x ∈ A. Obviously, Ae = top(P ) and Af = top(Q) are nonisomorphic simple left A/J-modules, and so ef = 0. Hence, ef ∈ J. Let 0 6= x ∈ P ∩ Q, so x = ae = bf. Note that xe = xf = x, and so x · (ef) = x. Therefore, x(ef)k = x for all k > 0. But since A is left k artinian, J is nilpotent, so (ef) = 0 for some k, and therefore x = 0, a contradiction. FROBENIUS-ARTIN ALGEBRAS AND INFINITE LINEAR CODES 9

Now, for each simple module T ∈ S, let P (T ) be its projective cover and Γ(T ) ⊆ S be set of simple modules that embed in P (T ). The next proposition shows that, under the extension property hypothesis, the socle of each P (T ) contains a single type of simple module, which is determined by T .

Proposition 3.4. Let A be a left artinian ring, which has the left extension property for simple modules. Then there is a permutation τ of the set S of simple left A-modules, such that for each S ∈ S, soc(P (T )) = τ(T )k(T ) for some k(T ).

Proof. By the above, we note that (Γ(T ))T ∈S are disjoint subsets of S. Moreover, obviously, F since A is left artinian, Γ(T ) is nonempty for all T . Thus, T ∈S Γ(T ) ⊆ S, and since S is ﬁnite, it follows that each Γ(T ) contains a single element, τ(T ) (i.e. Γ(T ) = {τ(T )}). Moreover, τ is injective, so it is a permutation (S is ﬁnite). The next (small step) is the following characterization of mininjectivity and the extension property for simple modules.

Proposition 3.5. Let A be a left artinian ring. (i) A is left mininjective if and only if for each left A-module S, the A-dual S∗ is simple or 0. (ii) If A has the left extension property for simple modules then for every simple left A-module S, its A-dual S∗ is a simple right A-module.

∗ Proof. (i) Let S ∈ S with S 6= 0. Let f, g ∈ HomA(S, A), S ∈ S, and assume f 6= 0, so f is injective. We have that the diagram below

f 0 / S / A

g ·a A is extended commutatively by some multiplication by a, if and only if f = g · a in S∗. Hence, the mininjectivity condition is equivalent to asking that S∗ is generated by any 0 6= f ∈ S∗, i.e. to S∗ is simple. (ii) Assume A is has the left extension property for simple modules. By the previous proposition, we have that each simple module embeds in some projective indecomposable left ∗ A-module (since τ is bijective), so S embeds in A. Therefore, S 6= 0 for S ∈ S. So far we have only used the extension property for simple modules. We note that the extension property of semisimple modules is potentially very strong:

Proposition 3.6. Let A be a left artinian ring which has the left extension property for semisimple modules. Then each principal projective indecomposable left A-module has simple socle (such a ring is called left QF-2).

Proof. Let Σ be the left socle of A, and for S ∈ S denote by ΣS the sum of its simple submodules isomorphic to S. Let f :Σ → A be a morphism of left modules. Obviously, f(Σ) ⊆ Σ. Q Note that EndA(Σ) is a semisimple artinian ring, with EndA(Σ) = MtS (End(S)), where S∈S ∼ P MtS (End(S)) = EndA(ΣS). Write f|Σ = fS with respect to this decomposition. If S has multiplicity one in Σ then fS is 0 or a monomorphism, so there is aS ∈ A such that 10 MIODRAG CRISTIAN IOVANOV fS(x) = x · aS, ∀ x ∈ S. Otherwise, since over a division ring, every matrix of order at least two is a sum of two invertible matrices by a result of Zelinsky [Z’54], we can write fS = gS +hS, for invertible gS, hS :ΣS → ΣS. Regarding gS, hS as monomorphisms ΣS → A, and applying the hypothesis, we see that they are given by right multiplication by suitable elements bS, cS. Therefore fS(x) = gS(s) + hS(x) = xbS + xcS = xaS for aS = bS + cS. Then we get P P f(x) = fS(x) = x( aS), so f is given by right multiplication. S S The above argument shows that the sequence 0 → (A/Σ)∗ → A∗ → Σ∗ → 0 is exact, and since Σ∗ is semisimple, we get an epimorphism A/J → Σ∗. Hence, length(Σ∗) ≤ length(A/J). At the same time, since by the previous proposition S∗ is simple for S ∈ S, we have ∗ lengthA(Σ ) = length(Σ) ≥ length(A/J), because each projective indecomposable has simple top, and some nontrivial socle. Hence, length(Σ) = length(A/J), and this implies that soc(P (S)) is simple for each simple left A-module S. The following extends the main result of [Ha’83], which is obtained for algebras over ﬁelds; it also extends the results of [W’08] on ﬁnite rings.

Theorem 3.7. Let A be an Artin algebra which is left QF-2 (i.e. left projective indecompos- ables have simple socle). The A is quasi-Frobenius. In particular, if A has the left extension property for semisimple modules, then A is quasi-Frobenius.

Proof. Consider B the basic algebra of A, which is Morita equivalent to A; thus B/Jac(B) is a product of skewfields. Obviously, the QF-2 property is Morita invariant, so B is QF-2. We first show that soc(P (S)) =6∼ soc(P (L)) if S =6∼ L, where P (S) is the projective cover of S. By contradiction , assume there are g : S → Be ⊂ B and h : S → Bf ⊂ nonzero morphisms with S simple and Be =6∼ Bf, for e, f primitive idempotents. Using the hypothesis, there are x, y ∈ B with g · x = h and h · y = g, so g(S) · xy = g(S). Hence, g(S) = g(S) · (xy)k for all k. Since g(S)e = g(S), h(S)f = h(S), we may assume x = exf. Now, x ∈ Jac(B) · Bf, since otherwise, since Bf is local, it would follow that Bex = Bf. But Bex is a quotient of Be (via right multiplication by x = xf), and since Bf is projective indecomposable, it would follow that Be =∼ Bf, a contradiction. In particular, x ∈ Jac(B), so g(S) = g(S) · (xy)k ⊆ J k for all k, which is not possible since J is nilpotent. As before, write τ(S) = soc(P (S)); then by the above τ is a permutation of S. Now, for each simple left B-module S, let P 0(S∨) be the projective cover of the right A-module S∨; then P 0(S∨)∨ is the injective hull of (S∨)∨ =∼ S. Since S is the (essential) socle of P (τ −1(S)), there is a monomorphism P (τ −1(S)) ,→ P 0(S∨)∨. Using that B is basic, we obtain a monomorphism M M M M B = P (S) = P (τ −1(S)) ,→ P 0(S∨)∨ = ( P 0(S∨))∨ = B∨ S∈S S∈S S∈S S∈S ∨ Since lK(B) = lK(B ), we must have an isomorphism above. Therefore, B is a Frobenius algebra, so it is quasi-Frobenius. Since quasi-Frobenius is Morita invariant, A is quasi- Frobenius too. The first part of statement (ii) in the following theorem is proved also in [NY’97, Corollary 4.8]; we provide an independent short proof. It is also a generalization of a result of Dieudoné [L, 16.2]. FROBENIUS-ARTIN ALGEBRAS AND INFINITE LINEAR CODES 11

Theorem 3.8. (i) Let A be a left artinian ring, which is left mininjective. Then A is right artinian too. (in particular, this holds when A has the left extension property for simple modules) (ii) If A is left artinian left and right mininjective, then A is QF. Consequently, if the left artinian ring A has the left and right extension property for simple linear codes, then A is QF .

Proof. (i) First note that for any left A-module of finite length X, its dual X∗ is of finite ∗ length and lengthA(X ) ≤ length(X). This can be easily shown by induction on the length of X. Indeed, consider an exact sequence 0 → S → X → N → 0, and dualize it to get the exact sequence 0 → N ∗ → X∗ → S∗. Since the morphism X∗ → S∗ can only be 0 or surjective (S∗ is simple or 0), in both cases X∗ has finite length if N ∗ does, and ∗ ∗ ∗ lengthA(X ) ≤ lengthA(N ) + lengthA(S ) ≤ lengthA(N) + lengthA(S) = lengthA(X). Moreover, note that if an equality holds for X, then backtracking, one sees that equalities ∗ hold for all subquotients of X. In particular, we see that AA = (AA) is of finite length, so A is also right artinian and length(AA) ≤ length(AA). (ii) Similarly as above, using the fact that the duals of right simple modules are simple or

0, we obtain length(AA) ≤ length(AA), so equality holds. As noted before, the equality ∗ ∗ length(A) = length((AA) ) implies that length(X) = length(X ) for every subquotient of A. In particular, for left submodules X ⊂ A, we see that the dual morphism A∗ → X∗ → 0 is surjective. But this shows precisely that Baer’s injectivity criterion is fulfilled for A, so A is left selfinjective. Hence, A is QF. As an alternative to the above finalization, we note that once we know that A is left and right artinian, since the duals of one-sided simple modules are simple or 0, by a result of Dieudoné [L, 16.2], the ring A is QF.

4. Compact groups and the combinatorics of the extension property In this section we prove that QF rings that are “essentially” infinite have the left and right extension property for linear codes. We will give two proofs of this main result. First, we need to recall some well known facts from the theory of compact groups. We refer the reader to [HM, St] for basic these facts on locally compact groups, and to [Ru] for the measure theory of locally compact spaces. Compact groups and Pontryagin duality. Let G be a locally compact group, i.e. a topological group whose topology is locally compact and Hausdorff separated. Then G has a left invariant sigma additive (Haar) measure m, unique up to multiplication by a scalar. If G is compact, then m(G) < ∞, and we convey to choose the measure with m(G) = 1. × Consider the group Hom(G, C ) of 1-dimensional characters of G, i.e. the 1-dimensional × ∼ × representations χ : G → C of G. Let T be the circle group; since T = C as groups, × we have an isomorphisms of groups Hom(G, C ) = Hom(G, T). By definition, for a locally compact group G, its dual group Gb consists of the continuous linear characters χ : G → T × of G. One advantage of considering T instead of C is its compact topology. Gb is again a group with operation given by pointwise multiplication of characters, and it has a topology, the topology of uniform convergence on compacts, equivalently, the compact-open topology on Gb as a subset of the space of continuous functions from G to T. This is the same as the G topology induced from T . When G is compact, this is the same as the uniform convergence 12 MIODRAG CRISTIAN IOVANOV on G. Note that Gb is a subset of the set of all representative functions on G (i.e. the space spanned by functions which occur in matrix representations ρ : G → Glm(C)). If G is an abelian locally compact group, then Gb is contains all the characters of G (i.e. it is “large”). Recall that Pontryagin duality asserts that the functor A −→ Ab is a self-inverse duality on the category of locally compact abelian groups, so A =∼ Abb for every locally compact abelian group A. The natural isomorphism is given by the evaluation map: x 7−→ ϕ(x) = (π 7→ π(x)). Under this duality, discrete groups (i.e. abelian groups endowed with the discrete topology) map to abelian compact groups and vice versa. We recall a few things on the Fourier analysis on compact groups. Let G be a locally compact abelian group and f ∈ L1(G), i.e. f is an integrable function on G. The Fourier transform of f is the function Ff = fb on Gb defined by Z fb(χ) = f(x)χ(x)dµ(x), G where µ is a fixed (left) Haar measure of G. fb is a continuous bounded function on Gb which “vanishes” at infinity. If ν is a Haar measure on Gb then the inverse Fourier transform of an L1 function g on Gb is the function F −1g given by Z gˇ = g(χ)χ(x)dν(x).

Gb The Fourier inversion formula says that ν can be chosen such that for every f ∈ L1(G), we have f = F −1fb almost everywhere in G, a formula which holds on all of G if f is continuous. We also recall that by the Peter-Weyl theorem, the set of characters of a compact abelian group G is dense in the set of all continuous functions on G (more generally, if G is not abelian, the algebra of representative functions on G is dense in the set of continuous functions on G). Furthermore, we will also use the fact that continuous functions on G are dense in L1(G) in the L1-norm (for example, by the more general Lusin’s theorem; [Ru]). We note the following well known fact concerning the integral of a character of a compact group: if χ is a character of the compact abelian group G, then R χ(x)dµ(x) = 0 if χ is G non-trivial character, and R χ(x)dµ(x) = µ(G) otherwise. Therefore, if (A, +) is an abelian G group, then Ab is compact and Z (1) π(x)dπ = δx,0,

Ab where we agree to ﬁx the (left and right) Haar measure m on Ab for which m(Ab) = 1. This is because the elements of A are identiﬁed with the characters of Ab by Pontryagin duality. Another way of seeing this is using the inverse Fourier transform. If 1 is the function Ab identically equal to 1 on Ab, and δ : A → C, δ(x) = δx,0 is the Kronecker delta function on A, then it is easy to see that F(δ) = δ = 1 , so F −1(1 ) = δ by the Fourier inversion formula, b Ab Ab which means precisely that R π(x)dπ = 0 if x 6= 0, and R π(x)dπ = 1 if x = 0. Ab Ab We will also need to use the following: if H is a subgroup of the compact group G with left

Haar measure µ and µ(H) > 0, then H is of ﬁnite index. Indeed, otherwise, if Hx1, . . . , Hxn FROBENIUS-ARTIN ALGEBRAS AND INFINITE LINEAR CODES 13

n F are distinct cosets of H, then µ(Hxi) = µ(H), so nµ(H) = µ( Hxi) ≤ µ(G), so n ≤ i=1 µ(G)/µ(H). Hence, there are at most µ(G)/µ(H) distinct cosets of H. Moreover, this shows 1 that in this situation, µ(H) = [G:H] µ(G).

Lemma 4.1. Let R be a QF ring, f, g : M → R be two morphisms such that ker(f) = ker(g). Then there is u ∈ U(R) such that f(x) = g(x)u for all x ∈ M.

Proof. Since ker(f) = ker(g) and R is selﬁnjective, there is a ∈ R such that f(x) = g(x)a for all x ∈ M. Let I = Im(g). Note that the map ϕa : I → R, ϕa(r) = ra (right multiplication by a) is injective; indeed, if g(x)a = 0 then f(x) = 0 so x ∈ ker(f) = ker(g) and thus g(x) = 0. Therefore, since R is left selﬁnjective, the diagram bellow is completed commutatively by some map ϕb = (−) · b · a 0 / I / R

⊆ · b R This shows that rab = r for r ∈ I, so 1 − ab ∈ r(I) = {t ∈ R|It = 0}. Hence, aR + r(I) = R, and since R is semilocal, by [B’64, Lemma 6.4] we have that (a+r(I))∩U(R) 6= ∅. Therefore, there is u ∈ U(R) such that I(u − a) = 0, so f(x) = g(x)a = g(x)u, ∀ x ∈ M, and the proof is ﬁnished. The following theorem generalizes the main result of [W’99]

Theorem 4.2. Let R = RF × RI be a ring with RF a ﬁnite Frobenius ring and RI a quasi- Frobenius ring with no nontrivial ﬁnite modules. Then R has the left (and right) extension property for linear codes.

Proof. Let λ, µ : L → Rn be two left R-module monomorphisms, such that wt(λ(x)) = wt(µ(x)) for all x ∈ L. If λi, µj are the components of λ and µ, this means that |{i|λi(x) 6= 0}| = |{j|µj(x) 6= 0}|. Using equation (1) above, this translates equivalently n n X Z X Z π(λi(x))dπ = π(µj(x))dπ i=1 j=1 Rb Rb where R is considered as the discrete abelian group (R, +). If λbi, µcj : Rb → Lb are the duals of λi, µj, and evx is the evaluation at x map on Lb, the formula above can be written n n P R P R evx ◦λbi = evx ◦µcj. Since this formula works for every x ∈ L, i.e. for every character i=1 j=1 Rb Rb evx of Lb and since set of characters of Lb is dense in the set of continuous functions on Lb, n n P R P R 1 we have f ◦ λbi = f ◦ µcj for f ∈ C(Lb), and then also for f in L (Lb) (since every i=1 j=1 Rb Rb f ∈ L1(Lb) is an L1-limit of continuous functions). One can also argue here by using the fact that every function in L1(Lb) (equivalently, every measurable function), is a pointwise almost everywhere limit of continuous functions. Hence, for f = χZ , the characteristic function of a n n P R P R measurable set Z ⊂ L, we have χZ ◦ λi = χZ ◦ µj, and since χZ ◦ µj = χ −1 , b b c c µj (Z) i=1 j=1 c Rb Rb 14 MIODRAG CRISTIAN IOVANOV this formula translates equivalently to n n X −1 X −1 m(λbi (Z)) = m(µcj (Z)) i=1 j=1 where m is the (left and right) normalized Haar measure of Rb. S −1 Let Z = Im(λbi). Since m(λbi (Z)) = 1 for all i, using the above formula, we get i −1 S −1 m(µcj (Z)) = 1, so m µcj (Im(λbi)) = 1 for each j. Fix some j, and let F be the i −1 set of those i’s for which m(µcj (Im(λbi))) > 0, we obviously have ! [ −1 m µcj (Im(λbi)) = 1. i∈F (we can live out the sets of measure 0). Note that Rb is also an R-bimodule, and each −1 µcj (Im(λbi)) is a right R-submodule of Rb = RcF ⊕ RcI , since one can easily see that λbi, µcj are right R-module morphisms. Also, since R = RF × RI , every R-module M decomposes as M = MF ⊕ MI , with MF an RF -module (annihilated by RI ) and MI an RI -module (i.e. −1 annihilated by RF ). The decomposition of Rb is Rb = RcF × RcI . Thus, µcj (Im(λbi)) = Xi ⊕ Yi −1 −1 for Xi ⊆ RcF , Yi ⊆ RcI . Moreover, since m(µcj (Im(λbi))) > 0, we see that µcj (Im(λbi)) is cofinite. Since RI has no finite nontrivial modules and RcI /Yi is finite, it follows that Yi = RcI . S −1 S Hence, µcj (Im(λbi)) = Xi × RcI . Note that Rb is also the product of RcF and RcI i∈F i∈F S −1 S as measure spaces, so 1 = m( µcj (Im(λbi))) = | Xi|/|RF | (since |RdF | = |RF |, because i∈F i∈F RF is finite, and m(RcI ) = 1). Therefore, we get that [ Xi = RcF . i∈F

Since RF is Frobenius, there is π0 ∈ RcF which generates RcF as left RF -module; but then −1 Π0 ∈ Xi for some i, so Xi = RcF , and therefore µcj (Im(λbi)) = Rb. Hence, Im(λbi) ⊆ Im(µcj), and, using the Peter-Weyl density, it is now easy to see that ker(µj) ⊆ ker(λi). We have shown that for each j there is i such that ker(µj) ⊆ ker(λi). Applying this several times (for both the λ’s and the µ’s), we can ﬁnd a sequence ker(µj1 ) ⊆ ker(λi1 ) ⊆ · · · ⊆ ker(µjt ) ⊆ ker(λit ) ⊆ .... Thus, we will have some repetitions, and therefore, we can ﬁnd some equality ker(λi) = ker(µj). Now Lemma 4.1 implies that λi(x) = µj(x) · u for some invertible u. Going back to the original hypothesis, this allows us to decrease the number of

λi’s (and µj’s) by one and apply induction on n. The case n = 1 is provided precisely by Lemma 4.1. A combinatorial proof. It may seem somewhat peculiar that the above algebraic problem requires measure theory and compact groups. We also provide a second purely combinatorial proof. We will need to use the following combinatorial result found, for example, in [P, Lemma 5.2] (see also [G’94]): FROBENIUS-ARTIN ALGEBRAS AND INFINITE LINEAR CODES 15

n S Lemma 4.3. Let A is an abelian group and A = Ai is a union of subgroups. Assume this i=1 S union is minimal, in the sense that none of the Ai’s can be eliminated (Ai 6⊆ Aj). Then j6=i n T A/ Ai is finite. i=1 Remark 4.4. In particular, we note that when M is a module with no nontrivial finite quo- tients and M = M1 ∪ · · · ∪ Mn, then M = Mi for some i. Indeed, otherwise if M 6= Mi for all i, then without loss of generality we can eliminate some of Mi’s and obtain a situation as T in the above Lemma; then M/ Mi is finite, so it must be trivial. Hence M = Mk for some i k, which is a contradiction.

With this we can prove:

Lemma 4.5. Let R be a ring which has no finite nontrivial modules. Let M be an R-module, n m P P and let (Ai)i=1,...,n, (Bj)j=1,...,m, be submodules of M such that 1Ai = 1Bj (here 1X is i=1 j=1 the characteristic function of the set X). Then n = m and the collection (A1,...,An) is a permutation of (B1,...,Bn). n n P P S S Proof. First note that the equality 1Ai = 1Bj implies that Ai = Bj, so Bj = i=1 j=1 i j S (Ai ∩ Bj) for every j. Since R has no finite nontrivial modules, using the above remark we i must have Bj = Ai ∩ Bj for some i depending on j, i.e. Bj ⊆ Ai. Similarly, we get that for each i there is j = j(i) with Ai ⊆ Bj. As before, we can find a sequence Ai1 ⊆ Bj1 ⊆ Ai2 ⊆

Bj2 ⊆ ...Aik ⊆ Bjk ⊆ ... , and since we have only ﬁnitely many Ai,Bj’s, we must have some repetition in Ai (and in Bj). In particular, we get that there is some equality Ai = Bj, and n n P P so 1Ai = 1Bj . Hence, the equality 1Ai = 1Bj reduces by one, and we can proceed by i=1 j=1 induction on the smallest of n and m. The case n = 1 (or m = 1) reduces similarly, and in that case it follows that m = 1 (respectively, n = 1) too. Hence, m = n and the conclusion follows. We can re-prove the result of this section:

Corollary 4.6. If R is a product of a ﬁnite Frobenius ring and a quasi-Frobenius ring with no nontrivial ﬁnite modules, then R has the left (and right) extension property for linear codes.

Proof. Let R = RF × RI be the stated decomposition. It is enough to show the property for each of RF and RI . For RF , the extension property holds by the ﬁnite version of Theorem n 4.2 or by the results of [W’99]. For RI , given λ, µ : L → R such that wt(λ(x)) = wt(µ(x)), then if λi, µi are the components of λ, µ and Ai = ker λi,Bi = ker(µi) then the condition n n P P translates to 1Ai = 1Bj . It follows from the above Lemma that for some permutation σ i=1 j=1 of (1, 2, . . . , n) we have ker(λi) = ker(µσ(i)). Lemma 4.1 implies again that there is ui ∈ U(R) such that λi(x) = µi(x) · ui, and the proof is ﬁnished. Finally, we can prove 16 MIODRAG CRISTIAN IOVANOV

Theorem 4.7. Let A be an Artin algebra. Then A has the extension property for linear codes if and only if A = AF ×AI with AF a ﬁnite ring and AI an Artin algebra with no ﬁnite nontrivial modules.

Proof. The if part follows from Theorem 4.2. For the only if part, first use Corollary 2.3 to note that A = AF × AI where AF is finite and AI has only infinite simple modules, so has no nontrivial finite modules. It is straightforward to see that the extension property induces to AF and AI . Therefore, AF is finite Frobenius and AI is quasi-Frobenius by Theorem 3.7. We end with a note on other possible extensions of results classical linear codes. The theory for codes over F2 carries over in good part to finite Frobenius rings. The classical MacWilliams identities [M] were extended by Wood in [W’99] to finite Frobenius rings. In view of the above interwtwining of the infinite theory with compact groups, it would perhaps be interesting, as a future direction of research, to generalize the MacWilliams identities to arbitrary Frobenius- Artin or quasi-Frobenius Artin algebras.

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Miodrag Cristian Iovanov University of Iowa Department of Mathematics, McLean Hall Iowa City, IA, USA and University of Bucharest, Faculty of Mathematics, Str. Academiei 14 RO-010014, Bucharest, Romania E–mail address: [email protected]; [email protected]