#### Preprint

<p>Matlis showed that the injective hull of a simple module over a commutative Noetherian ring is Artinian. Many non-commutative Noetherian rings whose injective hulls of simple modules are locally Artinian have been extensively studied recently. This property had been denoted by property (⋄). In this paper we investigate, which non-Noetherian semiprimary commutative quasi-local rings (<em>R</em>,<em>m</em>) satisfy property (⋄). For quasi-local rings (<em>R</em>,<em>m</em>) with <em>m</em>3=0, we prove a characterisation of this property in terms of the dual space of <em>S</em><em>o</em><em>c</em>(<em>R</em>). Furthermore, we show that (<em>R</em>,<em>m</em>) satisfies (⋄) if and only if its associated graded ring <em>g</em><em>r</em>(<em>R</em>) does.<br /> Given a field <em>F</em> and vector spaces <em>V</em> and <em>W</em> and a symmetric bilinear map <em>β</em>:<em>V</em>\texttimes<em>V</em>\textrightarrow<em>W</em> we consider commutative quasi-local rings of the form <em>F</em>\texttimes<em>V</em>\texttimes<em>W</em>, whose product is given by</p>

Paula Carvalho

Christian Lomp

Patrick Smith

### Publication

Year of publication: 2017