Rings whose modules have maximal submodules.

Faith, Carl

Publicacions Matemàtiques, Tome 39 (1995), p. 201-214 / Harvested from Biblioteca Digital de Matemáticas

Résumé

A ring R is a right max ring if every right module M ≠ 0 has at least one maximal submodule. It suffices to check for maximal submodules of a single module and its submodules in order to test for a max ring; namely, any cogenerating module E of mod-R; also it suffices to check the submodules of the injective hull E(V) of each simple module V (Theorem 1). Another test is transfinite nilpotence of the radical of E in the sense that radα E = 0; equivalently, there is an ordinal α such that radα(E(V)) = 0 for each simple module V. This holds iff each radβ(E(V)) has a maximal submodule, or is zero (Theorem 2). If follows that R is right max iff every nonzero (subdirectly irreducible) quasi-injective right R-module has a maximal submodule (Theorem 3.3). We characterize a right max ring R via the endomorphism ring Λ of any injective cogenerator E of mod-R; namely, Λ/L has a minimal submodule for any left ideal L = annΛM for a submodule (or subset) M ≠ 0 of E (Theorem 8.8). Then Λ/L0 has socle ≠ 0 for:

(1) any finitely generated left ideal L0 ≠ Λ; (2) each annihilator left ideal L0 ≠ Λ; and (3) each proper left ideal L0 = L + L', where L = annΛM as above (e.g. as in (2)) and L' finitely generated (Corollary 8.9A).

Détail

Publié le : 1995-01-01

MR MR1336364 | Zbl 0840.16002

DMLE-ID : 3755