A Note On Simple Modules Over Quasi-Local Rings

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 locally Artinian have been studied. This property had been denoted by property (lozenge). In this paper we investigate, which non-Noetherian semiprimary commutative quasi-local rings (R,m) satisfy property (lozenge). For quasi-local rings (R,m) with m(3) = 0, we prove a characterization of this property in terms of the dual space of Soc(R). Furthermore, we show that (R,m) satisfies (lozenge) if and only if its associated graded ring gr(R) does.Given a field F and vector spaces V and W and a symmetric bilinear map beta : V x V -> W we consider commutative quasi-local rings of the form F x V x W, whose product is given by(lambda(1), v(1),w(1))(lambda(2), v(2) ,w(2)) = (lambda(1)lambda(2), lambda(1)v(2) + lambda(2)v(1), lambda(1)w(2) + lambda(2)w(1) + beta(v(1), v(2)))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 satisfy (lozenge) if and only if it has a factor, whose associated graded ring is of the form F x V x F with V infinite dimensional and beta non-degenerated.