Π-Complemented Algebras Through Pseudocomplemented Lattices

Order (2012) 29:463–479 DOI 10.1007/s11083-011-9214-4

π-Complemented Algebras Through Pseudocomplemented Lattices

Juan Carlos Cabello · Miguel Cabrera · Antonio Fernández López

Received: 8 April 2010 / Accepted: 5 April 2011 / Published online: 5 May 2011 © Springer Science+Business Media B.V. 2011

Abstract For an ideal I of a nonassociative algebra A,theπ-closure of I is defined by I = Ann(Ann(I)), where Ann(I) denotes the annihilator of I, i.e., the largest ideal J of A such that IJ = JI = 0. An algebra A is said to be π-complemented if for every π-closed ideal U of A there exists a π-closed ideal V of A such that A = U ⊕ V. For instance, the centrally closed semiprime ring, and the AW∗-algebras (or more generally, boundedly centrally closed C∗-algebras) are π-complemented algebras. In this paper we develop a structure theory for π-complemented algebras by using and revisiting some results of the structure theory for pseudocomplemented lattices.

Keywords Pseudocomplemented lattices · Semiprime algebras · Closure operations · Complemented algebras · Decomposable algebras

Mathematics Subject Classifications (2010) Primary 17A60; Secondary 06D15

J.C. Cabello and M. Cabrera were supported by the MICINN and Fondos FEDER, MTM2009-12067, and, in addition, by the Junta de Andalucía Grant FQM290. A. Fernández López was supported by the MEC and Fondos FEDER, MTM2007-61978 and MTM2010-19482. J. C. Cabello · M. Cabrera Departamento de Análisis Matemático, Facultad de Ciencias, Universidad de Granada, 18071 Granada, Spain J. C. Cabello e-mail: [email protected] M. Cabrera e-mail: [email protected]

A. Fernández López (B) Departamento de Álgebra, Geometría y Topología, Facultad de Ciencias, Universidad de Málaga, 29071 Málaga, Spain e-mail: [email protected] 464 Order (2012) 29:463–479

1 Introduction

Unless otherwise stated, A will denote a not necessarily associative algebra over an arbitrary commutative ring of scalars. Recall that A is said to be complemented if for every ideal I of A there exists an ideal J of A such that A = I ⊕ J. Examples of complemented algebras are the null algebras and the decomposable algebras, where by a null algebra we mean an algebra with zero product, a decomposable algebra is an algebra that is isomorphic to a direct sum of simple algebras, and a simple algebra is a non-null algebra lacking nonzero proper ideals. By regarding any algebra as a left module over its multiplication algebra, the standard characterization of completely reducible modules can be rewritten as follows: A non-null algebra is complemented if, and only if it is the direct sum of a null algebra and a decomposable algebra. Given an algebra A and a closure operation ∼ on the complete lattice IA of all (two-sided) ideals of A, borrowing terminology from the theory of Banach spaces, we will say that a ∼-closed ideal U of A is ∼-complemented (resp. ∼-quasicomplemented)inA if there exists a ∼-closed ideal V of A,calleda∼-complement (resp. ∼-quasicomplement)ofU, such that A = U ⊕ V (resp. A = (U ⊕ V) ∼ ). We say that A is a ∼-complemented (resp. ∼-quasicomplemented) algebra when every ∼-closed ideal of A is ∼-complemented (resp. ∼-quasicomplemented) in A. Note that if ∼ is the discrete closure, i.e., U = U for all U ∈ IA, then complementarity, ∼-complementarity and ∼-quasicomplementarity agree. A natural task is the description of the lattice of the ∼-closed ideals. Thus, the study of the lattice of all .-closed ideals and the complementarity in such a lattice has been widely discussed in specific Banach algebras: algebras of functions, and particularly algebras of sequences, (see, e.g., [7, Chapter 4] and [37]); and operator algebras (see, e.g., [30]and[33]). It is worth pointing out that .-quasicomplemented algebras are generalized annihilator normed algebras, which were introduced and studied by B. Yood in the associative context [38], by the third author and A. Rodríguez in the nonassociative setting [15], and revisited in [4]. In the latter paper, the ε-closure was introduced, and proved that the ε-quasicomplemented algebras with zero annihilator are precisely the multiplicativatively semiprime algebras. More recently, in [5], a structure theory for ε-complemented algebras has been achieved. In the present paper we study those algebras which are complemented with respect to the π-closure, i.e., the closure defined on IA by the annihilator operator Ann(.). For each ideal I of A,Ann(I) denotes the largest ideal J of A such that = = π = ( ( )) Iπ IJ JI 0,andthe -closure of I is defined by I Ann Ann I .Theset A of all π-closed ideals of A is a complete lattice for the meet and join operations given by Ui = Ui and Ui = Ui. In section one the concept of annihilator operator on a semilattice is introduced, and a generalization to semilattices with an annihilator operator of a celebrated theorem of Frink and Glivenko [19] for pseudocomplemented semilattices is proved. As a consequence, it is shown that the π-quasicomplemented algebras are precisely the semiprime algebras. In section two we provide a π-structure theory for semiprime algebras: (i) every semiprime algebra is a subdirect product of two semiprime Order (2012) 29:463–479 465 algebras, one of which is π-radical, i.e., it does not contain minimal π-closed ideals, and the other is π-decomposable, i.e., it is the π-closure of the sum of its minimal π-closed ideals, and (ii) π-decomposable semiprime algebras are characterized as essential subdirect products of prime algebras, equivalently, as those semiprime algebras such that the Boolean algebra of their π-closed ideals is atomic; in particular, semiprime algebras satisfying the chain condition on annihilator ideals are π- decomposable. A list of examples of π-decomposable algebras, which runs through different classes of nonassociative algebras, is given at the end of this section. In section three we discuss pseudocomplemented lattices with additive closure. In section four a structure theory for π-complemented algebras is developed: (i) an algebra is π-complemented if and only it is semiprime and its π-closure is additive, (ii) every π- complemented algebra is the direct sum of a π-radical π-complemented algebra and a π-decomposable π-complemented algebra, (iii) an algebra with zero annihilator coincides with its π-socle if, and only if, it is a direct sum of prime algebras, and (iv) the π-decomposable π-complemented algebras are the hereditary subdirect products of prime algebras. Examples of π-decomposable π-complemented algebras in both functional analysis and pure algebra are collected in this section. Finally, we show that the algebra of all Lebesgue measurable functions on the unit interval is π-radical and π-complemented.

2 Pseudocomplemented Semilattices and Semiprime Algebras

We refer to [19], [22]or[16], for basic results on pseudocomplemented semilattices and Boolean algebras. Let (S;∧, 0) be a (meet) semilattice with a least element. By an annihilator operator on S we mean an unary operation x → x∗ of S into itself satisfying the following conditions for all x, y ∈ S (1) x ≤ y ⇒ y∗ ≤ x∗, (2) x ≤ x∗∗, (3) x ∧ y = 0 ⇒ y ≤ x∗. Conditions (1) and (2) say that any annihilator operator defines a symmetric Galois connection.Hence ∗ ∗ ∗ (4) If {xα}⊆S and α xα exists, then α xα exists and α xα = α xα , (5) x∗ = x∗∗∗, and the mapping x → x := x∗∗ is a closure operation on S. By property (5), x ∈ S is closed if, and only if, x = y∗ for some y ∈ S. Denote by B(S) the set of all closed elements of S (also called the skeleton of S [22]). Since x ∧ 0 = 0 for all x ∈ S, it follows from (3) that x ≤ 0∗,so (6) 1 := 0∗ is the largest element both S and B(S). As usual we consider the binary operation ∇:S × S → B(S) defined by

x∇ y := (x∗ ∧ y∗)∗.

Proposition 2.1 Let (S;∧, 0,∗ ) be a semilattice with annihilator operator. Then (B(S);∇) is a join semilattice with 1 = 0∗ as a largest element and 1∗ as a least element. 466 Order (2012) 29:463–479

Moreover, if S is a lattice, then x∇ y = x ∨ y for all elements x, yand(B(S);∧, ∇) is a lattice.

Proof Check that (x, y) → x∇ y defines a join map in B(S). Moreover, if S is a lattice, we have by (4) that x∇ y = x ∨ y and u ∧ v = (u∗ ∨ v∗)∗ ∈ B(S) for all x, y ∈ S and u,v ∈ B(S).

Recall that a pseudocomplemented semilattice is a semilattice with a least element (S;∧, 0) such that for each x ∈ S there exists a largest element x∗ such that x ∧ x∗ = 0. It follows from the basic properties of the pseucomplementation [19]that the pseudocomplemented semilattices are precisely the semilattices with annihilator operator (S;∧, 0,∗ ) such that x ∧ x∗ = 0 for all x ∈ S. A celebrated theorem of Frink and Glivenko proves that if (S;∧, 0,∗ ) is a pseudocomplemented semilattice, then (B(S);∧, ∇,∗ , 0, 1) is a Boolean algebra, i.e., a distributive complemented lattice, with the Boolean complement of an element being its pseudocomplement. This result was proved for complete distributive lattices by V. Glivenko in his early work [20], and was later refined by O. Frink in [17]. For a brief proof of this result, see [27]. Next we will prove an extension of the theorem of Frink and Glivenko for semilattices S with annihilator operator which yields complementation in B(S), but before we show the relationship between pseudocomplemented semilattices and semilattices with an annihilator operator in the lattice of the ideals of a nonassociative algebra. In what follows, by an algebra we will mean a not necessarily associative algebra over an arbitrary but f ixed ring of scalars.

Example 2.2 For a (two-sided) ideal I of an algebra A, denote by Ann(I) the annihilator of I, i.e., the largest ideal J of A such that IJ = JI = 0. It is easy to verify that the mapping I → Ann(I) defines an annihilator operator in the complete lattice IA of all ideals of A. The associated closure will be called the π-closure of A. Note that A is semiprime, i.e., I2 = 0 implies I = 0 for any ideal I of A,if, and only if, the annihilator operator makes IA into a pseudocomplemented lattice. Nevertheless, an algebra might be not semiprime and yet the lattice of its ideals be pseudocomplemented. Take for instance Z12.

Theorem 2.3 Let (S;∧, 0,∗ ) be a semilattice with annihilator operator and let (B(S);∇) be the join semilattice of its closed elements. Then the following assertions are equivalent: (i) For any u ∈ B(S), there exists v ∈ B(S) such that u ∧ v = 0 and u∇v = 1. (ii) x → x∗ is a pseudocomplementation on S. (iii) (B(S);∧, ∇,∗ , 0, 1) is a Boolean algebra.

Proof (i) ⇒ (ii). First we show that 1∗ = 0.Since1∗ is closed, there exists v ∈ B(S) such that 1∗ ∧ v = 0 and 1∗∇v = 1. Therefore 1 = 1∗∇v = (1 ∧ v∗)∗ = v∗∗ = v. Then 1∗ = 1∗ ∧ 1 = 1∗ ∧ v = 0. Order (2012) 29:463–479 467

Given u ∈ B(S),letv ∈ B(S) be such that u ∧ v = 0 and u∇v = 1.By(3),v ≤ u∗,so u∇u∗ = 1. Then, by what we have just proved,

0 = 1∗ = (u∇u∗)∗ = u∗ ∧ u.

Therefore, u ∧ u∗ = 0 for any closed element u of S.Ifx ∈ S is not necessarily closed, then x ∧ x∗ ≤ x ∧ x∗ = 0, and consequently x ∧ x∗ = 0. Thus the annihilator operator defines a pseudocomplementation. The implication (ii) ⇒ (iii) is the well-known result due to Frink and Glivenko [19, Theorem 1], while the implication (iii) ⇒ (i) follows from the very definition of Boolean algebra.

Iπ π From now on we will denote by A the complete lattice of all -closed ideals of an algebra A. Together Theorem 2.3 and Example 2.2 yield the following

Corollary 2.4 For an algebra A the following assertions are equivalent: Iπ (i) A is a complemented lattice. (ii) Aissemiprime. (iii) I → Ann(I) is a pseudocomplementation on IA. Iπ (iv) A is a Boolean algebra.

The following two properties of the pseudocomplementation [19] will be often used throughout the paper. (7) (x ∧ y)∗ = (x ∧ y)∗. (8) 1∗ = 0. Moreover, (9) (x ∧ u)∗ ∧ u = x∗ ∧ u,foranyx ∈ S and u ∈ B(S). Indeed, taking into account (7) and using the distributivity of the meet with the ∇- operation, we see that

(x ∧ u)∗ ∧ u = (x ∧ u)∗ ∧ u = (x∗∇u∗) ∧ u = x∗ ∧ u.

Proposition 2.5 Let (S;∧, 0,∗ ) be a pseudocomplemented semilattice and let u,v ∈ B(S) with u ≤ v.Then (i) The interval [u,v] is a pseudocomplemented semilattice with pseudocomplementation given by x → x⊥ := (x ∧ u∗)∗ ∧ v, and with closure operator being the restriction of the closure operator on S; hence B([u,v]) =[u,v]∩B(S). (ii) The Boolean algebras B([0,v]) and B([v∗, 1]) are isomorphic. ∗ (iii) B(S) ={w1 ∧ w2 : w1 ∈ B([u, 1]), w2 ∈ B([u , 1])}.

Proof (i) Let x ∈[u,v].Sincex ∧ u∗ ≤ u∗ it follows that u = u ≤ (x ∧ u∗)∗, and consequently x⊥ = (x ∧ u∗)∗ ∧ v ∈[u,v]. Moreover, note that

x ∧ x⊥ ∧ u∗ ≤ x ∧ (x ∧ u∗)∗ ∧ u∗ = 0, 468 Order (2012) 29:463–479

hence x ∧ x⊥ ≤ u = u,andsox ∧ x⊥ = u. On the other hand, for any z ∈[u,v] such that x ∧ z = u we have x ∧ z ∧ u∗ = 0,hencez ≤ (x ∧ u∗)∗,andso z ≤ (x ∧ u∗)∗ ∧ v = x⊥. Summarizing, x⊥ is the pseudocomplement of x in [u,v]. By applying (9) we see that

(x ∧ u∗)∗ ∧ u∗ = x∗ ∧ u∗ = x∗ and (x∗ ∧ v)∗ ∧ v = x∗∗ ∧ v = x∗∗.

Hence ⊥ ∗ x⊥⊥ = (x ∧ u∗)∗ ∧ v = (x ∧ u∗)∗ ∧ v ∧ u∗ ∧ v = (x∗ ∧ v)∗ ∧ v = x∗∗,

that is, B([u,v]) =[u,v]∩B(S). (ii) A natural anti-isomorphism of B([0,v]) onto B([v∗, 1]) is given by x → x∗, but Boolean algebras are dual. (iii) For w ∈ B(S), we see that u∇w ∈ B([u, 1]), u∗∇w ∈ B([u∗, 1]) and ∗ ∗ (u∇w) ∧ (u ∇w) = (u ∧ u )∇w = w. Conversely, if w1 ∈ B([u, 1]) and ∗ w2 ∈ B([u , 1]),thenclearlyw1 ∧ w2 ∈ B(S).

3 π-Structure Theory of Semiprime Algebras

Semiprime algebras are here studied through the Boolean algebra of its π-closed ideals. The reader is referred to Lambek’s book [29, p. 111] where the Boolean algebra of a semiprime ring is used in Goldie ring theory. Recall that an ideal D of an algebra A is said to be essential if for every nonzero ideal I of A we have I ∩ D = 0.IfA is semiprime, then I is essential if, and only if, Ann(I) = 0.

Lemma 3.1 Let A be a semiprime algebra and D be an essential ideal of A. Then Ann(I ∩ D) = Ann(I) for every ideal I of A.

Proof Clearly Ann(I) ⊆ Ann(I ∩ D).SinceA is semiprime we see that I ∩ D ∩ Ann(I ∩ D) = 0, and since D is an essential ideal of A we deduce that I ∩ Ann(I ∩ D) = 0. Therefore, Ann(I ∩ D) ⊆ Ann(I).

A subdirect product of a (nonempty) family {Aλ}λ∈ of nonzero algebras is any subalgebra A of the full direct product Aλ such that the canonical projections pλ : A → Aλ are onto. A subdirect product is said to be essential if it contains an essential ideal of the full direct product. The next proposition records some basic results on subdirect products of algebras and describes the structure of their π-closed ideals. To save notation, given a subdirect product A ≤ Aλ of algebras, whenever an Aλ is contained in A, it will be regarded as an ideal of A.

Proposition 3.2 Let A be a subdirect product of a family {Aλ}λ∈ of algebras. Then

(i) A is semiprime if all the Aλ are semiprime. ( ) = ∩ ( ( )) (ii) For any ideal I of A, Ann I A AnnAλ pλ I . π π I ⊆{ ∩ λ : λ ∈ I ,λ∈ }. (iii) A A U U Aλ Order (2012) 29:463–479 469

(iv) If A is an essential subdirect product, then A is semiprime if, and only if, each Aλ is semiprime. In this case, A contains an essential ideal Dλ of Aλ for each index λ,and

π π I ={ ∩ λ : λ ∈ I ,λ∈ }. A A U U Aλ

(v) Conversely, if all the Aλ have zero annihilator and A contains an essential ideal Dλ of Aλ for each index λ, then A is actually an essential subdirect product.

Proof

(i) Since pλ : A → Aλ is onto, pλ(I) is an ideal of Aλ for each ideal I of A.If 2 2 I = 0 then pλ(I) = 0 and hence pλ(I) = 0 for all λ,soI = 0 and A is semiprime. ⊆ ( ) ( ) ⊆ ( ( )) (ii) For any I and J ideals of A, J Ann I if, and only if, pλ J AnnAλ pλ I λ ( ) = ∩ ( ( )) for each .HenceAnn I A AnnAλ pλ I . (iii) It follows from (ii). (iv) If each Aλ is semiprime, then A is semiprime by (i). Suppose that A is semiprime and that it contains an essential ideal D of Aλ.LetJλ be an 2 ideal of Aλ (regarded as an ideal of Aλ). If Jλ = 0 then Jλ ∩ D = 0 by semiprimeness of A and hence Jλ = 0 because D is an essential ideal, which proves that Aλ is semiprime. Now set Dλ := D ∩ Aλ for each index λ.IfJλ is a nonzero ideal of Aλ,thenJλ ∩ Dλ = Jλ ∩ D = 0 because D is essential, which proves that Dλ is an essential ideal of Aλ, contained in A. Moreover, we have by (ii) that any π-closed ideal U of A is of the required form. Suppose = ∩ = ( ) conversely that U A Uλ,whereUλ AnnAλ Jλ ,withJλ being an ideal of Aλ for each index λ.SinceDλ = D ∩ Aλ is an essential ideal of Aλ,we = ( ∩ ) ⊆ have by Lemma 3.1 that U λ AnnAλ Jλ Dλ . Thus we may suppose Jλ A for each λ.SetJ := A ∩ Jλ.ThenJλ = pλ(J) for each λ, and hence, by (ii), U = Ann(J), as required. (v) Suppose that each Aλ has zero annihilator and that A contains an essential ideal Dλ of Aλ for each index λ.LetJ be a nonzero ideal of Aλ and let λ ∈ ( ) = ( ) = = ( ) = be such that pλ J 0.SinceAnnAλ Aλ 0, JAλ pλ J Aλ 0 or Aλ J = 0.Inanycase,J ∩ Aλ is a nonzero ideal of Aλ and hence

J ∩ Dλ = (J ∩ Aλ) ∩ Dλ = 0

because Dλ is essential. This proves that Dλ is an essential ideal of Aλ, contained in A, as required.

Recall that an algebra A is said to be non-null if A2 = 0,andprime if IJ = 0, I, J ideals of A, implies I = 0 or J = 0.

Proposition 3.3 Let A be a non-null algebra. Then A is prime if, and only if, Iπ ={ , } A 0 A .

Proof If A is prime, then Ann(A) = 0 and Ann(I) = 0 for any nonzero ideal I of Iπ ={ , } Iπ ={ , } ( ) = 2 = A,so A 0 A . Conversely, if A 0 A , then Ann A 0, because A 0. Moreover, A is semiprime: I2 = 0, I an ideal of A, implies I ⊆ Ann(I) and hence 470 Order (2012) 29:463–479

Ann(I) = A if I is nonzero, which leads to a contradiction. Now IJ = 0, I and J ideals of A, implies I ∩ J = 0 by semiprimeness, and hence J ⊆ Ann(I).ThenJ = 0 or Ann(I) = A. If the latter, I ⊆ Ann(Ann(I)) = Ann(A) = 0, which proves that A is prime.

By a minimal π-closed ideal of an algebra A we mean an atom of the complete Iπ lattice A. Let U ⊆ V be π-closed ideals of a semiprime algebra A satisfying U ⊆ V.We set Bπ ([U, V]) to denote the Boolean algebra of all π-closed ideals W such that U ⊆ W ⊆ V. Proposition 3.4 Let A be a semiprime algebra and let U be a π-closed ideal of A. Then / π Iπ (i) A U is semiprime and the Boolean algebras of -closed ideals A/U , Bπ ([U, A]) and Bπ ([0, Ann(U)]) are isomorphic. (ii) If U is nonzero, then U is a minimal π-closed ideal of A if, and only if, A/Ann(U) is prime.

Proof (i) The quotient map I → q(I) defines a lattice isomorphism of [U, A] onto 2 2 IA/U .NowI ⊆ U implies (I ∩ Ann(U)) ⊆ U ∩ Ann(U) = 0 and hence I ∩ Ann(U) = 0 by semiprimeness of A.ThenI ⊆ U = U, which proves that A/U is semiprime, equivalently, by Corollary 2.4, q(I) → AnnA/U (q(I)) is a pseudocomplementation on A/U. Since the interval [U, A] inherits pseudocomplementation from IA by Proposition 2.5, it follows from the uniqueness of Iπ Bπ ([ , ]) the pseudocomplementation that the Boolean algebras A/U , U A and Bπ ([0, Ann(U)]) are isomorphic. (ii) If U = 0, then Ann(U) = A,andso,by(i),A/Ann(U) is a nonzero semiprime algebra. Hence, by Proposition 3.3, A/Ann(U) is prime if, and only if, Iπ ={ , / ( )} Iπ A/Ann(U) 0 A Ann U . But, again by (i), the Boolean algebra A/Ann(U) is isomorphic to Bπ ([0, U]). Thus A/Ann(U) is prime if, and only if, U is a minimal π-closed ideal.

Deﬁnition 3.5 Let A be an algebra. We set π−Soc(A) to denote the π-socle of A, i.e. the sum of its minimal π-closed ideals. Then the π-radical of A is defined by π−Rad(A) := Ann(π−Soc(A)). An algebra A will be called π-decomposable if π−Soc(A) is π-dense in A (i.e., A = π−Soc(A)), and A will be called π-radical if it does not contain minimal π- closed ideals.

Proposition 3.6 Any semiprime algebra A is an essential subdirect product of a π- decomposable semiprime algebra Ad and a π-radical semiprime algebra Ar.Infact, we can take Ad = A/π−Rad(A) and Ar = A/π−Soc(A). Moreover, Iπ ={ ∩ : ∈ Bπ ([π− ( ), ]), ∈ Bπ ([π− ( ), ])}. A U V U Soc A A V Rad A A

Proof Set Ad := A/π−Rad(A) and Ar := A/π−Soc(A). By Proposition 3.4(i), both two algebras are semiprime. Moreover, Iπ =∼ Bπ ([0, π−Soc(A)]),soA is Ad d Order (2012) 29:463–479 471

π-decomposable, and Iπ =∼ Bπ ([0,π−Rad(A)]),soA is π-radical. Since π−Soc(A) Ar r and π−Rad(A) can be regarded as essential ideals of Ad and Ar respectively and π−Rad(A) ∩ π−Soc(A) = 0, A is an essential subdirect product of Ad by Ar.The Iπ description of A follows from Proposition 2.5(iii).

Theorem 3.7 For a nonzero algebra A with zero annihilator the following conditions are equivalent: (i) Aisπ-decomposable. Iπ (ii) A is semiprime and the Boolean algebra A is atomic. (iii) A is an essential subdirect product of a family of nonzero prime algebras. Iπ P( ) In this case, A is isomorphic to the power set X , where X is the set of the minimal π-closed ideals of A, i.e., any nonzero π-closed ideal of A is the π-closure of the set of all minimal π-closed ideals of A contained in it.

Proof

(i) ⇒ (ii) Let U be a π-closed ideal of A and let {Uλ : λ ∈ } the family of the minimal π-closed ideals of A.IfU does not contain any Uλ, then, by minimality of Uλ, Uλ ∩ U = 0,andsoU ⊆ Ann(Uλ) for all λ.Then U ⊆ Ann(Uλ) = Ann Uλ = Ann(A) = 0, λ∈ λ∈ = Iπ and so U 0. Thus the complete lattice A is atomic. 2 To prove that A issemiprimewebeginbynotingthatUλ = 0 for all λ ∈ . Indeed, if there exists λ ∈ such that U 2 = 0,then 0 λ0 ⊆ = = ( ) = , Uλ0 Ann Uλ Ann Uλ Ann A 0 λ∈ λ∈ = and so Uλ0 0, which is a contradiction. Now, assume the existence of a nonzero ideal I of A such that I2 = 0.ThenI ⊆ Ann(I) and hence ⊆ ( ) 2 = Iπ λ ∈ I Ann I , equivalently, I 0.Since A is atomic, there exists 0 2 such that Uλ ⊆ I, which again leads to a contradiction since U = 0. 0 λ0 Thus A is semiprime. ⇒ { } π Iπ (ii) (iii) Let Uλ λ∈ be the set of all minimal -closed ideals of A.Since A is atomic, Ann(Uλ) = 0, which implies that A is a subdirect product of the family of algebras {Aλ},withAλ := A/Ann(Uλ) for each λ.By Proposition 3.4(ii), each Aλ is prime, and since Uλ can be regarded as an essential ideal of Aλ, it follows from Proposition 3.2(v) that A is an essential subdirect product of the Aλ. (iii) ⇒ (i) Suppose that A ≤ Aλ is an essential subdirect product of a family of nonzero prime algebras. Taking into account Propositions 3.2(iv) and 3.3, we have

Iπ = ∩ : ⊆ . A A Aλ λ∈ 472 Order (2012) 29:463–479

Therefore, {A ∩ Aλ : λ ∈ } is the family of all minimal π-closed ∩ ideals of A. For a fixed subset of , it is clear that A λ∈ Aλ π ( ∩ ) is the smallest -closed ideal of A containing λ∈ A Aλ ,andso λ∈(A ∩ Aλ) = A ∩ λ∈ Aλ. Thus A is π-decomposable. Iπ P( ) That in this case A is isomorphic to the power set X ,whereX is the set of the minimal π-closed ideals of A, is a standard result of lattice theory.

Note that by (5) the two chain conditions on annihilators, i.e., π-closed ideals, are equivalent. If this is the case for an algebra A, we will simply say that A satisfies the chain condition on annihilators. As a consequence of [16, Theorem 3.7], we obtain the following corollary of Theorem 3.7.

Corollary 3.8 For a semiprime algebra A the following statements are equivalent: (i) A satisf ies the chain condition on annihilators. (ii) A does not contain inf inite direct sums of nonzero ideals. (iii) A is an essential subdirect product of f initely many prime algebras. Iπ (iv) The Boolean algebra A is f inite. In this case, the uniform (two-sided) dimension of A coincides with the (f inite) cardinal of the set of its minimal π-closed ideals, and it is equal to the annihilator length of A.

The reader is referred to [13, Theorem 4.1 and Corollary 4.2] for a former version of Theorem 3.7 and Corollary 3.8, and to [28, Theorem 11.43] for a related result for semiprime rings. The following list provided examples of semiprime algebras which are π- decomposable, some of them actually satisfy the chain condition on annihilators. Semiprime Goldie rings, semiprime rings with essential socle and Fountain-Gould orders in semiprime rings with minimal condition on principal right sided ideals [18]. Nondegenerate Goldie alternative rings [9] and Fountain-Gould left orders in nondegenerate alternative rings which coincide with their socle [21]. Semiprime Goldie Jordan algebras [39], classical orders in nondegenerate unital Jordan algebras with finite capacity [14], nondegenerate noncommutative Jordan algebras with essential socle [11] and nondegenerate Jordan algebras satisfying local Goldie conditions [12]. Nondegenerate Lie algebras with essential socle, in particular, nondegenerate Artinian Lie algebras [8].

4 Pseudocomplemented Lattices with Additive Closure

Deﬁnition 4.1 A closure operation x → x in a join semilattice (S;∨) will be called additive if x ∨ y = x ∨ y for all x, y ∈ S.

Proposition 4.2 Let (L;∧, ∨,∗ , 0, 1) be a pseudocomplemented lattice, and let (B(L);∧, ∇,∗ , 0, 1) be its Boolean algebra. Then the following conditions are equivalent: (i) For any u,v ∈ B(L) such that u ∧ v = 0,u∨ v ∈ B(L). Order (2012) 29:463–479 473

(ii) For any u,v ∈ B(L),u∨ v ∈ B(L). (iii) For any x, y ∈ L, (x ∧ y)∗ = x∗ ∨ y∗. (iv) The closure operator x → x = x∗∗ is additive. In this case, L satisf ies the Stone identity: 1 = x ∨ x∗ for any x ∈ L.

Proof (i) ⇒ (ii) Let u,v ∈ B(L). By Theorem 2.3, 1 = u∇u∗ and

v = v ∧ 1 = v ∧ (u∇u∗) = (v ∧ u)∇(v ∧ u∗) = (v ∧ u) ∨ (v ∧ u∗)

since (v ∧ u) ∧ (v ∧ u∗) = 0 and both v ∧ u and v ∧ u∗ are closed. Then u ∨ v = u ∨ (v ∧ u∗) is closed. (ii) ⇒ (iii) Let x, y ∈ L.Thenx∗ ∨ y∗ = (x∗ ∨ y∗)∗∗ = (x ∧ y)∗ = (x ∧ y)∗. (iii) ⇒ (iv) Let x, y ∈ L.Thenx ∨ y = x∗∗ ∨ y∗∗ = (x∗ ∧ y∗)∗ = (x ∨ y)∗∗ = x ∨ y. (iv) ⇒ (ii) For any u,v ∈ B(L), u ∨ v = u ∨ v = u ∨ v ∈ B(L). Suppose that these equivalent conditions hold, then for any ∗ x ∈ L, 1 = x ∨ x∗ = x ∨ x∗ since x is closed and x = x∗.

Proposition 4.3 Let (L;∧, ∨,∗ , 0, 1) be a lattice with annihilator operator, and let B(L) its skeleton. If L is modular, then the following conditions are equivalent: (i) For any x ∈ L there exists v ∈ B(L) such that 1 = x ∨ v and x ∧ v = 0. (ii) (L;∧, ∨,∗ , 0, 1) is a lattice with pseudocomplementation and additive closure.

Proof (i) ⇒ (ii) It follows from Theorem 2.3 that L is pseudocomplemented, and 1 = u ∨ u∗ for any u ∈ B(L).Letv ∈ B(L) be such that u ∧ v = 0.Use modularity and the distributivity of the meet with respect to ∇-jointoget

u ∨ v = (u ∨ v) ∧ (u ∨ u∗) = u ∨ ((u∇v) ∧ u∗) = u ∨ (v ∧ u∗) = u ∨ v

since v ≤ u∗. Hence the closure is additive by Proposition 4.2. (ii) ⇒ (i) By Theorem 2.3, for any x ∈ L, 1 = x ∨ x∗ = x ∨ x∗, since the closure is additive.

5 π-Complemented Algebras

Deﬁnition 5.1 An algebra A is said to be π-complemented if for any π-closed ideal U of A there exists a π-closed ideal V such that A = U ⊕ V.

Examples of π-complemented Banach algebras are the AW∗-algebras, or more generally, the boundedly centrally closed C∗-algebras, [2, Example 3.3.1(2) and Re- mark 3.2.7]. Another examples of π-complemented associative algebras are provided by the rings of quotients of a semiprime ring [3]. More generally, we have the following result.

Proposition 5.2 Every centrally closed semiprime ring R is π-complemented. 474 Order (2012) 29:463–479

Proof Let I be an ideal of R.By[3, Theorem 2.3.9(i)] there exists a unique idempotent E(I) in the extended centroid C of R such that

(1 − E(I))C ={c ∈ C : cI = 0}.

Since (1 − E(I))I = 0, it follows that (1 − E(I))R ⊆ Ann(I). Conversely, since IRAnn(I) = 0, it follows that E(I)Ann(I) = 0 because of [3, Lemma 2.3.10], and hence Ann(I) ⊆ (1 − E(I))R. Therefore Ann(I) = (1 − E(I))R, and thus I = E(I)R is a π-complemented ideal of R.

Proposition 5.3 For an algebra A the following conditions are equivalent: (i) Aisπ-complemented, (ii) A is semiprime and its π-closure is additive. In this case, any π-closed ideal U of A is a π-complemented algebra and Iπ ={ ⊕ : ∈ Iπ , ∈ Iπ }. A V W V U W Ann(U)

Proof The lattice of ideals of A is modular, and, by Corollary 2.4, A is semiprime if, and only if, the annihilator operator defines a pseudocomplementation. Apply Proposition 4.3 to get the equivalence of the two conditions. Let U be a π-closed ideal of A.SinceA = U ⊕ Ann(U), any ideal of U is an ideal of A. Moreover, by Proposition 2.5, the π-closure operator of IU is the restriction I I π Iπ to U of that of A.HenceU is a -complemented algebra. The description of A follows then from Proposition 3.2(iv).

Proposition 5.4 Let A ≤ Aλ be a subdirect product of algebras such that A contains the direct sum of all the Aλ.

(i) If A is π-complemented, then all the Aλ are π -complemented. (ii) If all the Aλ are π-complemented and A = Aλ or A = Aλ,thenAisπ- complemented.

Proof

(i) Since each Aλ can be regarded as a π-closed ideal of A, Proposition 3.2(iv), Aλ is π-complemented by Proposition 5.3. (ii) By Proposition 3.2(iv), the π -closed ideals of Aλ ( Aλ respectively) are of the form U = Uλ (U = Uλ respectively) where each Uλ is a π-closed ideal of Aλ. If each Aλ is π-complemented, then Aλ = U λ ⊕ Vλ.Taking V := Vλ (V = Vλ respectively), we get Aλ = U ⊕ V ( Aλ = U ⊕ V respectively). Thus, in both cases, A is π-complemented.

Proposition 3.6 can be refined for π-complemented algebras as follows.

Proposition 5.5 For every algebra A the following assertions are equivalent:

(i) Aisπ-complemented. (ii) A is isomorphic to Ar Ad,whereAr is a π-radical π-complemented algebra and Ad is a π-decomposable π-complemented algebra. Order (2012) 29:463–479 475

In this case, Ar = π−Rad(A) and Ad = π−Soc(A).

Proof (i) ⇒ (ii) By definition,

A = π−Soc(A) ⊕ Ann(π−Soc(A)) = π−Soc(A) ⊕ π−Rad(A)

and both π−Soc(A) and π−Rad(A) are π-complemented algebras by Proposition 5.4(i). Moreover, since any minimal π-closed ideal of A is a minimal π-closed ideal of π−Soc(A), π−Soc(A) is π-decomposable and π−Rad(A) = Ann(π−Soc(A)) is π -radical. (ii) ⇒ (i) Suppose conversely that A = Ar Ad,whereAr is a π-radical π- complemented algebra, and Ad is a π-decomposable π-complemented algebra. By Proposition 5.4(ii), A is π-complemented, and by Propo- sition 5.3, Iπ ={U ⊕ V : U ∈ Iπ , V ∈ Iπ }, which also proves that A Ar Ad Ad = π−Soc(A) and Ar = π−Rad(A).

The simplest examples of π-decomposable π-complemented algebra are furnished by those algebras A such that A = π−Soc(A).

Proposition 5.6 For a nonzero algebra A with zero annihilator the following assertions are equivalent: (i) A = π−Soc(A). (ii) A is isomorphic to a direct sum of a family of nonzero prime algebras. Iπ ={ : ⊆ } { } π (iii) A λ∈ Uλ ,where Uλ λ∈ is the family of all minimal -closed ideals of A. In this case, A is a π-decomposable π-complemented algebra.

Proof

(i) ⇒ (ii) By Theorem 3.7, A is semiprime. Since A = Uλ is the direct sum of its minimal π-closed ideals, then A = U ⊕ Ann(U) for any minimal π-closed ideal of A, and hence U =∼ A/Ann(U) is a prime algebra by Proposition 3.4(ii). (ii) ⇒ (iii) Let A = Aλ a direct sum of nonzero prime algebras. By Proposition π I ={ , λ} λ 3.3, Aλ 0 A for each . Now it follows from Proposition 3.2(iv) π I ={ λ : ⊆ } that ⊕λ∈ Aλ λ∈ A . (iii) ⇒ (i) This implication is clear. Finally, assume that A is an algebra satisfying the equivalent conditions in the statement. From (iii) it is clear that A is π-decomposable and π- complemented.

Next we describe the π-decomposable π-complemented algebras in terms of subdirect products, and then we will give an example of a π-radical π-complemented algebra. Let {Aλ}λ∈ be a family of algebras. For each subset of the index set ,set p : Aλ → Aλ to denote the homomorphism defined by the conditions 476 Order (2012) 29:463–479 pλ p = pλ for any λ ∈ ,andpλ p = 0 otherwise. Note that each p is a projection, with p\ being its complementary projection.

Deﬁnition 5.7 A subdirect product A ≤ λ∈ Aλ of algebras is said to be hereditary if p(A) ⊆ A for any subset of .

Full direct products, direct sums, and algebras of bounded sequences and of sequences converging to zero are examples of hereditary subdirect products.

Proposition 5.8 Let A ≤ Aλ be a subdirect product of a family {Aλ}λ∈ of nonzero prime algebras. Iπ ={ ( ) ∩ : ⊆ } (i) If the subdirect product is essential, then A p A A .Moreover, Ann(p(A) ∩ A) = p\(A) ∩ Aforall ⊆ . (ii) If the subdirect product is hereditary, it is an essential subdirect product, in fact, ⊆ Iπ ={ ( ) : ⊆ } Aλ A, and A p A .

Proof

(i) By Proposition 3.2(iv), any π-closed ideal U of A is of the form A ∩ Uλ, where the Uλ are π-closed ideals of Aλ. Since each Aλ is a prime algebra, by Proposition 3.3, either Uλ = 0 or Uλ = Aλ for each index λ.Set ={λ ∈ : Uλ = Aλ}.ThenU = p(A) ∩ A and Ann(U) = p\(A) ∩ A. (ii) Since Aλ = pλ(A) ⊆ A for each index λ, we have Aλ ⊆ A.ThenA ≤ Aλ is an essential subdirec product and hence we have by (i) that the π-closed ideals of A are of the form p(A), ⊆ .

Theorem 5.9 A nonzero algebra is π-complemented and π-decomposable if, and only if, it is a hereditary subdirect product of nonzero prime algebras.

Proof Suppose that A is π-complemented and π-decomposable. Then A is semiprime by Proposition 5.3, and hence A is an essential subdirect product of a family {Aλ} of nonzero prime algebras by Theorem 3.7. We claim that A is actually a hereditary subdirect product. Given a subset of the index set , we have by Proposition 5.8(i) that U := p(A) ∩ A is a π-closed ideal of A,withAnn(U ) = p\(A) ∩ A. Then p(U) = U and p\(Ann(U)) = Ann(U).NowA = U Ann(U) implies p(A) = p(U) = U since p and p\ are complementary projections, which proves that A ≤ Aλ is a hereditary subdirect product, as claimed. Suppose conversely that A is a hereditary subdirect product of a family {Aλ} of nonzero prime algebras. Then λ∈ Aλ ⊆ A by Proposition 5.8(ii), so A is π- decomposable by Theorem 3.7. Let U be a π-closed ideal of A.ThenU = p(A) and Ann(U) = p\(A) for some subset of the index set by Proposition 5.8(ii), and since p and p\ are complementary projections, A = p(A) p\(A) = U Ann(U),soA is π-complemented, as required.

Corollary 5.10 The following (real or complex) algebras are examples of π- decomposable π-complemented algebras:

(1) The algebra c0 of all null sequences. Order (2012) 29:463–479 477

(2) The algebra p (1 ≤ p < ∞) of all absolutely p-summable sequences. (3) The algebra ∞ of all bounded sequences.

Other Banach algebras which are π-complemented and π-decomposable are the dual B∗-algebras of Kaplansky [26], and the proper H∗-algebras of Ambrose [1]. These classical Banach algebras are examples of the so-called complemented Banach algebras [36] introduced by Tomiuk. In any semiprime complemented Banach algebra A a (two-sided) ideal I of A is closed (for the norm topology) if, and only if, it is π-closed. In this case, A = I ⊕ Ann(I). The reader is referred to [15]forsome nonassociative extensions of dual B∗-algebras and proper H∗-algebras, which are also π-complemented and π-decomposable. Further examples of π-complemented and π-decomposable algebras, now in the purely algebraic context, are recorded in the following list: Nondegenerate Jordan algebras with descending chain condition on principal inner ideals [31], in particular, with minimum condition [25] and complemented Jordan algebras [32]. Nondegenerate Lie algebras coinciding with their socle [8], and complemented Lie algebras [10]. Finite dimensional algebras under suitable notion of semisimplicity, for instance, algebras with a nondegenerate symmetric asociative bilinear form [34, Theorem 2.6], in particular, semisimple alternative algebras [34, Theorem 3.12], semisimple Jordan algebras [34, Corollary 4.6], semisimple noncommutative Jordan algebras [34, Corollary 5.5], semisimple structurable algebras [35, Theorem 7], semisimple Lie algebras [24, Structure theorem on p. 71], and algebras with semiprime multiplication algebra ([23], [17], or [4, Proposition 3.1]). Now give the announced example of a π-radical π-complemented algebra. Since finite dimensional algebras contain minimal π-closed ideals, such an algebra must be infinite dimensional. In fact, our example is an algebra of measurable functions. For the definition and properties of the Lebesgue measure we refer the reader to the classical book [6] and merely establish some notation. We denote by M the σ -algebra of all Lebesgue measurable subsets of the unit interval [0, 1], and by λ : M →[0, 1] the Lebesgue measure. We start with the following elemental fact.

∗ Lemma 5.11 For each subset A of [0, 1],thereexistA∗, A ∈ M such that ∗ ∗ A∗ ⊆ A ⊆ A and λ(E\A∗) = λ(A \F) = 0,forallE, F ∈ M with E ⊆ A ⊆ F.

Proof Set

α = inf{λ(F\E) : E, F ∈ M such that E ⊆ A ⊆ F} ∈N , ∈ M ⊆ ⊆ λ( \ ) ≤ α+ 1 and, for each n , choose En Fn such thatEn A Fn and Fn En n . = ∗ = , ∗ ∈ M Now, consider the sets A∗ n∈N En and A n∈N Fn, and note that A∗ A ∗ ∗ and A∗ ⊆ A ⊆ A . Moreover, since A \A∗ ⊆ Fn\En for every n, it follows that

∗ 1 α ≤ λ(A \A∗) ≤ λ(F \E ) ≤ α + , n n n ∗ and as a result λ(A \A∗) = α. Given E, F ∈ M with E ⊆ A ⊆ F, note that ∗ ∗ ∗ A∗ ∪ E ⊆ A ⊆ A ∩ F and (A ∩ F)\(A∗ ∪ E) ⊆ A \A∗, and hence we get that ∗ λ((A ∩ F)\(A∗ ∪ E)) = α. Moreover, by considering the decomposition

∗ ∗ ∗ A \A∗ = (E\A∗) ∪[(A ∩ F)\(A∗ ∪ E)]∪(A \F), 478 Order (2012) 29:463–479 we see that ∗ ∗ ∗ α = λ(A \A∗) = λ(E\A∗) + λ((A ∩ F)\(A∗ ∪ E)) + λ(A \F)

∗ = λ(E\A∗) + α + λ(A \F), ∗ hence λ(E\A∗) = λ(A \F) = 0, and the proof is complete.

Example 5.12 The algebra M of all equivalence classes (under equality almost everywhere) of Lebesgue measurable functions on [0, 1] with pointwise operations is a π-radical π-complemented algebra. Clearly M is a commutative associative algebra with a unit 1. Given E ∈ M, note that the characteristic function χE is an idempotent of M, χEχEc = 0,and c 1 = χE + χEc ,whereE denotes the complement of E in [0, 1]. Therefore

M = χEM ⊕ χEc M, π and hence χEM ∈ IM and Ann(χEM) = χEc M. We claim that π IM ={χEM : E ∈ M}.

Given f ∈ M, consider the measurable set S f := {x ∈[0, 1]: f (x) = 0}, and note = χ χ = that f S f f and S f fg,whereg is the measurable function defined by ( ) = ( )−1 ∈ ( ) = M = χ M g x f x if x S f and g x 0 otherwise. Therefore f S f . Note that, for a given π-closed ideal U of M, we have = ( ( )) = χ M U Ann Ann U Ann S f f∈Ann(U) = Ann(χ M) = χ c M. S f S f f∈Ann(U) f∈Ann(U) Therefore Iπ = χ M :{ : ∈ }⊆M . M Ei Ei i I i∈I { : ∈ }⊆M χ M = χ M Given a family Ei i I , it is immediately verified that i∈I Ei E , = ( ) where E i∈I Ei ∗, and so we have proved our claim. Now it follows from M = χEM ⊕ χEc M that M is π-complemented. Finally, note that χEM = 0 if, and only if, λ(E) = 0. Therefore, if U is a nonzero π-closedideal,andU = χEM for suitable E ∈ M, then by choosing F ⊆ E with 0 <λ(F)<λ(E) we see that χF M is a nonzero π-closed ideal of M strictly contained in U. Thus M is π-radical.

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