A NOTE ON SIMPLE MODULES OVER QUASI-LOCAL RINGS

Abstract

Matlis showed that the injective hull of a simple module over

a commutative Noetherian ring is Artinian. In several recent papers, non-

commutative Noetherian rings whose injective hulls of simple modules are lo-

cally Artinian have been studied. This property had been denoted by property

(). In this paper we investigate, which non-Noetherian semiprimary commu-

tative quasi-local rings (R;m) satisfy property (). For quasi-local rings (R;m)

with m3 = 0, we prove a characterization of this property in terms of the dual

space of Soc(R). Furthermore, we show that (R;m) satises () if and only if

its associated graded ring gr(R) does.

Given a eld F and vector spaces V and W and a symmetric bilinear

map : V V ! W we consider commutative quasi-local rings of the form

F V W, whose product is given by

(1; v1;w1)(2; v2;w2) = (12; 1v2 + 2v1; 1w2 + 2w1 + (v1; v2))

in order to build new examples and to illustrate our theory. In particular we

prove that a quasi-local commutative ring with radical cube-zero does not sat-

isfy () if and only if it has a factor, whose associated graded ring is of the

form F V F with V innite dimensional and non-degenerated.

Keywords

• Quasi-local ring,injective hull,simple module,niteness condi- tion

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