Sufficient conditions for the existence of a left quotient ring of a ring decomposed into a direct sum of left ideals

S. L. Berlov

Zapiski Nauchnykh Seminarov POMI, Problems in the theory of representations of algebras and groups. Part 4, Tome 227 (1995), pp. 9-14

Résumé

Let $R=P_1\oplus P_2\oplus\dots\oplus P_n$ be a decomposition of a ring into a direct sum of indecomposable left ideals. Assume that these ideals possess the following properties: (1) any nonzero homomorphisms $\varphi\colon P_i\to P_j$ is a monomorphism; (2) if subideals $Q_1,Q_2$ of the ideal $P_j$ are isomorphic to the ideal $P_i$, then there exists a subideal $Q_3\subseteq Q_1\cap Q_2$, which is also isomorphic to $P_i$. It is proved that, under these asumptions, a left quotient ring of the ring $R$ exists. This left quotient ring inherits properties (1), (2) and satisfies condition (3): any nonzero homomorphism $\varphi\colon P_i\to P_i$ is an automorphism of the ideal $P_i$. Bibliography: 2 titles.