Socle chains of abelian regular semiartinian rings

On socle chains of semiartinian rings with primitive factors artinian

“…factors modulo annihilator of a simple module, are artinian, reflect the structure of single semisimple slices. The notion is introduced in [14] and structural theory of dimension sequences is developed in [18]. The aim of this paper is to generalize results on dimension sequences for a suitable subclass of regular right semiartinian rings R with primitive factors artinian, which is possible for those R satisfying the condition that every ideal finitely generated as two-sided ideal is finitely generated as right ideal.…”

“…Finally, i(I) ≤ sup α<λ (n α ) ≤ i(K) by Lemma 2.1. The following example shows that we cannot work with sets of central idempotents in a general ring R ∈ R as in the case of abelian regular rings in [18] Example 2.3. Let F be a field and put P = ω × ω.…”

“…Obviously, ghR is generated by an idempotent. P Let us prove a new version of [18, Lemma 1.4] which serves as a basic tool for improving results of [18]. Lemma 2.6.…”

“…Since every ideal contained in Soc(R) is generated by central idempotents, (1) and (2) are easy generalization of [18,Lemma 1.4].…”

“…Thus E(eR ∩ Soc(R)), and so E(eR) is infinite by (1). P Since neither all idempotents nor all central idempotents replace in the general case role of (central) idempotents in abelian regular semiartinian rings as it is illustrated by Examples 2.3, 2.4, we need formulate a new version of [18,Lemma 3.2]. Lemma 2.7.…”