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 ${\text{End}}_R(S)\cong Q\prime /I,$ where Q′ is a suitable subring of maximal right ring of quotients of R and $I=\{r\in Q\prime |rS=0\}$ 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 $L_K(E)$ with essential socle is socle-injective if and only if $L_K(E)$ 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.

Keywords:

• Arbitrary graph
• essential ideal
• Leavitt path algebras
• socle-injective rings

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 16D50
• 16D60

Acknowledgments

The authors would like to thank Professor Juana Sánchez Ortega for organising the CIMPA research school on Topics in Ring Theory-2018 which took place in Muizenberg (South Africa), where the initial discussion of this project started. Also, the authors would like to sincerely thank the referee for valuable notes and comments which greatly improved this article. Finally, the second author thanks the editor Professor Alberto Facchini for his help and guidance during the review process.

Additional information

Funding

The third and fourth authors were partially supported by the Junta de Andalucia and Fondos FEDER, jointly, through projects FQM-336, FQM-7156, UMA18-FEDERJA-119 and also by the Spanish Ministerio de Economia y Competitividad and Fondos FEDER, through project MTM2016-76327-C3-1-P, and by the Spanish Ministerio de Ciencia e Innovación through project PID2019-104236GB-I00.