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. ∼-quasicomplemen- ted)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 complementar- ity, ∼-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 partic- ular, 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 sec- tion 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 ele- ment (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.