Bulletin of theKorean Mathematical SocietyBKMS

ISSN(Print) 1015-8634 ISSN(Online) 2234-3016

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Bull. Korean Math. Soc.

Published online July 1, 2022

A note on Artinian local rings

Kui Hu, Hwankoo Kim, and Dechuan Zhou

Southwest University of Science and Technology, Hoseo University

Abstract

In this note, we prove that an Artinian local ring is G-semisimple (resp., SG-semisimple, 2-SG-semisimple) if and only if its maximal ideal is G-projective (resp., SG-projective, 2-SG-projective). As a corollary, we obtain the global statement of the above. We also give some examples of local G-semisimple rings whose maximal ideals are $n$-generated for some positive integer $n$.

Keywords: Artinian local ring; QF-ring; G-semisimple ring; SG-semisimple ring; 2-SG-semisimple ring.

MSC numbers: 13G05; 13D03