# Rings whose modules have maximal submodules.

Publicacions Matemàtiques (1995)

- Volume: 39, Issue: 1, page 201-214
- ISSN: 0214-1493

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topFaith, 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 -

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