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Generalized primary rings and ideals. (English) Zbl 1105.16002

An ideal \(I\) of the associative ring \(R\) is said to be a generalized right primary (g.r.p.) ideal if whenever \(A\) and \(B\) are ideals of \(R\) such that \(AB\subseteq I\), then either \(A\subseteq I\) or \(B^n\subseteq I\) for some \(n\). The ring \(R\) is called a g.r.p. ring if the zero ideal of \(R\) is a g.r.p. ideal. When these conditions are satisfied for principal ideals \(A\) and \(B\), then \(I\) is said to be a principal generalized right primary (p.g.r.p.) ideal and, respectively, \(R\) is said to be a p.g.r.p. ring. Analogously the authors define generalized left primary (g.l.p.) ideals and rings, and principal generalized left primary (p.g.l.p.) ideals and rings.
Examples are given showing that p.g.r.p. (p.g.l.p.) does not imply g.r.p. (g.l.p.) even for commutative rings. Examples are also given to illustrate that the conditions are indeed one-sided. Conditions are given for the intersection of generalized primary ideals to be generalized primary ideals. Ascending chain conditions on ideals are used in this context. Various set inclusion relations and permutation identities are used to establish conditions for one-sided generalized primary conditions to be two-sided.

MSC:

16D25 Ideals in associative algebras
16D70 Structure and classification for modules, bimodules and ideals (except as in 16Gxx), direct sum decomposition and cancellation in associative algebras)
16N40 Nil and nilpotent radicals, sets, ideals, associative rings
16P70 Chain conditions on other classes of submodules, ideals, subrings, etc.; coherence (associative rings and algebras)
16U80 Generalizations of commutativity (associative rings and algebras)
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