
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 = fv1; : : : ; vng 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 jGj, 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 Poincar´e 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 cryptog- raphy. 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 par- tial 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 × A1, with Af finite Frobenius and A1 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 × A1 with Af a finite Frobenius ring and A1 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 × A1 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).
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