Generalized Munn rings

• Published in: Open mathematics (Warsaw, Poland) Vol. 20; no. 1; pp. 1066 - 1081
• Main Authors: Guo, Junying, Guo, Xiaojiang
• Format: Journal Article
• Language: English
• Published: Warsaw De Gruyter 06.10.2022; De Gruyter Poland
• Subjects: 05C25; 15A30; 16S36; generalized matrix ring; generalized Munn ring; generalized path algebra; Matrices (mathematics); Rings (mathematics); semiprime ring; semiprimitive ring
• ISSN: 2391-5455; 2391-5455
• DOI: 10.1515/math-2022-0487

Generalized Munn rings exist extensively in the theory of rings. The aim of this note is to answer when a generalized Munn ring is primitive (semiprimitive, semiprime and prime, respectively). Sufficient and necessary conditions are obtained for a generalized Munn ring with a regular sandwich matrix to be primitive (semiprimitive, semiprime and prime, respectively). Also, we obtain sufficient and necessary conditions for a Munn ring over principal ideal domains to be prime (semiprime, respectively). Our results can be regarded as the generalizations of the famous result in the theory of rings that for a ring , is primitive (semiprimitive and semiprime, respectively) if and only if so is . As applications of our results, we consider the primeness and the primitivity of generalized matrix rings and generalized path algebras. In particular, it is proved that a path algebra is a semiprime if and only if it is semiprimitive.