Rings whose modules have maximal submodules.

Faith, Carl. "Rings whose modules have maximal submodules.." Publicacions Matemàtiques 39.1 (1995): 201-214. <http://eudml.org/doc/41212>.

@article{Faith1995,
abstract = {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).},
author = {Faith, Carl},
journal = {Publicacions Matemàtiques},
keywords = {Teoría de anillos; Teorema módulo máximo; simple right modules; quasi-injective right modules; right max rings; Hamsher modules; right perfect rings; maximal submodules; injective hulls; injective cogenerators; endomorphism rings; left annihilator ideals},
language = {eng},
number = {1},
pages = {201-214},
title = {Rings whose modules have maximal submodules.},
url = {http://eudml.org/doc/41212},
volume = {39},
year = {1995},
}

TY - JOUR
AU - Faith, Carl
TI - Rings whose modules have maximal submodules.
JO - Publicacions Matemàtiques
PY - 1995
VL - 39
IS - 1
SP - 201
EP - 214
AB - 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).
LA - eng
KW - Teoría de anillos; Teorema módulo máximo; simple right modules; quasi-injective right modules; right max rings; Hamsher modules; right perfect rings; maximal submodules; injective hulls; injective cogenerators; endomorphism rings; left annihilator ideals
UR - http://eudml.org/doc/41212
ER -