INICIO | 03 de marzo de 2024

# Rings whose modules have maximal submodules.

Título inglés Rings whose modules have maximal submodules. Anillos cuyos módulos tienen submodulos máximos. Faith, Carl Dep. Math. Rutgers Univ., New Brunswick (New Jersey), Estados Unidos 0214-1493 1995, 39 (1): 201-214, 23 Ref. articulo Inglés 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). 120105 Teoría de anillos ; Teorema módulo máximo MR1336364 Acceso al artículo completo
Equipo DML-E
Instituto de Ciencias Matemáticas (ICMAT - CSIC)
rmm()icmat.es