A question on $*$-regular rings
Bull. Korean Math. Soc. 2018 Vol. 55, No. 5, 1333-1338
Published online September 1, 2018
Jian Cui, Xiaobin Yin
Anhui Normal University, Anhui Normal University
Abstract : A $*$-ring $R$ is called $*$-regular if every principal one-sided ideal of $R$ is generated by a projection. In this note, several characterizations of $*$-regular rings are provided. In particular, it is shown that a matrix ring $M_n(R)$ is $*$-regular if and only if $R$ is regular and $1+x_1^*x_1+\cdots+x_{n-1}^*x_{n-1}$ is a unit for all $x_i$ of $R;$ which answers a question raised in the literature recently.
Keywords : $*$-regular ring, regular ring, matrix ring, GN property
MSC numbers : 16E50, 16W10
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