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

Abstract

Let R=P1⊕P2⊕…⊕Pn 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 ϕ: Pi→Pj is a monomorphism; (2) if subideals Q1, Q2 of the ideal Pj are isomorphic to the ideal Pi, then there exists a subideal Q3⊆Q1⊎Q2, which is also isomorphic to Pi. 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 ϕ: Pi→Pi is an automorphism of the ideal Pi. Bibliography: 2 titles.