Primary orders with a~finite numbers of indecomposable representations

Yu. A. Drozd ; V. V. Kirichenko

Izvestiya. Mathematics , Tome 7 (1973) no. 4, pp. 711-732

Résumé

Let $\Lambda$ be a semisimple $Z$-ring and $C$ its center. Assume that for any prime ideal $\mathfrak p\subset C$ the ring $\Lambda_{\mathfrak p}$ is primary. Let $\overline\Lambda$ be the intersection of the maximal over-rings of $\Lambda$, $I=\overline\Lambda/\Lambda$ and $I'=\operatorname{rad}I$. We prove that $\Lambda$ has a finite number of indecomposable integral representations if and only if $\overline\Lambda$ is a hereditary ring, $I$ has two generators as a $\Lambda$-module, and $I'$ is cyclic.