Structure theory for a generalised Boolean ring

Abstract

Structure theory for a generalised Boolean ringBy N.V. SUBRAHMANYAM in Waltair, IndiaTo Professor V. Ramaswami § 0. Introduction 0.0: Let R be any (associative) ring, in which PI: each element has a minimal idempotent duplicator in the centre of R; i.e. if a ~/t, there exists an idempotent a ° in the centre of R such that (i) aa ° = a, and (ii) for every idempotent e which satisfies ae = ea ---a, must aOe = a °.Such a ring R seems to be a natural generalisation for a Boolean ring (in terms of its lattice structure) and it includes all the known generalisations for {[3]) Boolean rings, the regular rings of vo~ N:~N~ with identity and without non-zero nilpotent elements etc., and further the so-called assoc~te rings of Ittv~G SUSSMA~ [3] which embrace all these classes of rings and also the biregular rings of AR~NS and KAPL/~NSK¥ etc. For these rings, the author has extended [2] the lattice theory [3] of SUSSMAN for associate rings and has also shown, by means of examples, that a Pl-ring (i,e. a ring with property P1) actually belongs to a wider class than the class of associate rings; and it was shown in [2] that a Pl-ring (and hence an associate ring) is composed of certain maximal sets, which are distributive lattices (Boolean rings), all isomorphic to one another: precisely, we have proved (see theorem 5 [2]) that each element of a Pl-ring is contained in a maximal set which is lattice isomorphic to the set R 0 of idempotents in it; and we were tempted to conjecture (which we shall disprove in this paper) to the effect that all the maximal sets are lattice isomorphic to one another (see Cor. 10.2 [2]). Further, a Pl-ring was shown to be isomorphic to a subdireet sum of subdireetly irreducible rings, each with unity element and (so) without other non-zero idempotent elements. 0.1: In the present paper, we examine the principal ideal structure of a Pl-ring satisfying the atomisti~ condition (see below for definition) and show that such a ring is isomorphic to a "special" subdirect sum of the ideals generated by its atomic idempotents, with further specialisations under additional conditions. These results extend those of IRVING Suss~z~ [4] for associate rings (to Pl.rings) which were obtained under the comp/ete a~m/~/c c~nd/tion and thus present improvements even in this case; also we shall show that many of our structure theorems cannot be improved. Finally, we obtain conditions on a Pl-ring which are necessary to render it an associate ring, and also present, in the last section, an example which seems to suggest certain directions in which the author's lattice theory for Prrings may be extended.Math. Ann. 141 21

Almost distributive lattices

“…Among them regular rings (Von Neumann (1936)), /7-rings (McCoy and Montgomery (1937)), biregular rings (Arens and Kaplansky (1948)), associate rings (Sussman (1958)) and Parings (Subrahmanyam (1960a)) are worth mentioning. It is observed, in Maddana Swamy and Manikyamba (to appear), that the class of associate rings coincides with the classes of well known Baer rings and w-domain rings (Subrahmanyam (1960b)). Also, it coincides with *-rings (Saracino and Weisfenning (1975)).…”

Closure Operators on Complete Almost Distributive Lattices-III

“…In [17] Swamy and Rao introduced the concept of an Almost Distributive Lattice (ADL) as a common abstraction of almost all the existing ring theoretic generalizations of a Boolean algebra like p-rings [12], regular rings [11], biregular rings [16], associate rings [10], p 1 -rings [13], triple systems [15], baer rings [1], m-domain rings [14] and * -rings [2] on one hand and the class of distributive lattices on the other. Thus, a study of any concept in the class of ADLs will yield results in all the classes of algebras mentioned above.…”

Boolean semirings

“…Further, the structure theorems of Boolean rings have been extended (see [2] and also [4]) to Pl-rings, where we have studied the structure of a Pl-ring in terms of its "idem-ideals" (=principal ideals generated by idempotent elements) under various conditions: atomlsticity, complete atomisticity, etc. Further, the structure theorems of Boolean rings have been extended (see [2] and also [4]) to Pl-rings, where we have studied the structure of a Pl-ring in terms of its "idem-ideals" (=principal ideals generated by idempotent elements) under various conditions: atomlsticity, complete atomisticity, etc.…”

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Introduction

A ring R, in which every element is idempotent, is called a "Boolean ring"; and it is well-known that a Boolean ring (with unity element) is also a distributive (and complemented) lattice.

Several generalisations for a Boolean ring are known --p-rings, regular rings, biregular rings, associate rings [3], Pl-rings [I], etc., of which, the class of associate rings contains, as special cases, those of p-rings, regular rings with unity element and without nonzero nilpotent elements etc. Further, the structure theorems of Boolean rings have been extended (see [2] and also [4]) to Pl-rings, where we have studied the structure of a Pl-ring in terms of its "idem-ideals" (=principal ideals generated by idempotent elements) under various conditions: atomlsticity, complete atomisticity, etc. Further, the structure theorems of Boolean rings have been extended (see [2] and also [4]) to Pl-rings, where we have studied the structure of a Pl-ring in terms of its "idem-ideals" (=principal ideals generated by idempotent elements) under various conditions: atomlsticity, complete atomisticity, etc.

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