Socle-injective semiprime rings, with some applications to Leavitt path algebras

Abstract

A ring R is called right (or left) socle-injective if every R-homomorphism from the right (or left) socle of R into R extends to R. In this article, we show that any semiprime ring R with socle S, is socle-injective if and only if where Q′ is a suitable subring of maximal right ring of quotients of R and is an ideal of Q′. Furthermore, an explicit structure of the ring Q′ is presented for a semiprime socle- injective ring, with essential socle. As an application, we show that a unital Leavitt path algebra with essential socle is socle-injective if and only if is semisimple, hence von Neumann regular. Moreover, we observed that socle-injective Leavitt path algebras are left-right symmetric. We also have provided examples to illustrate our results.