Bulletin of the
Korean Mathematical Society BKMS

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: Primary 13G05, 13D03

Supported by: The second author was supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education (2021R1I1A3047469). The third author was supported by National Natural Science Foundation of China (12101515).