A new duality theorem for semisimple modules and characterization of Villamayor rings

Abstract

Abstract.We prove the theorem: If R is a ring whose right ideals satisfy the double annihilator condition with respect to a semisimple right Ä-module W , then every right ideal is an intersection of maximal right ideals, consequently R is a right V (for Villamayor) ring, and W is then necessarily a cogenerator of mod-R . (The converse is well known.) We use this to give a new proof of a theorem of ours on right Johns rings.

Modules With Annihilator Conditions

“…This condition was termed by Faith-Menal [15] as saying that M satisfies the double annihilator condition with respect to right ideals. For a subset K of N and a subset X of Hom R ðN; MÞ, put AnðKÞ ¼ ff : f 2 Hom R ðN; MÞ and fðKÞ ¼ 0g and KeðXÞ ¼ \fkerðgÞ : g 2 Xg.…”

“…It was proved in [15] that R is a right V-ring if and only if R is M-annular for some semisimple module M, and that in this case M is a cogenerator in Mod-R. When R R is a V-module, we call R a right V-ring.…”