A QUESTION ON ⁎-REGULAR RINGS

A ${\ast}-ring$ R is called ${\ast}-regular$ if every principal one-sided ideal of R is generated by a projection. In this note, several characterizations of ${\ast}-regular$ rings are provided. In particular, it is shown that a matrix ring $M_n(R)$ is ${\ast}-regular$ if and only if R is regular and $1+x^*_1x_1+{\cdots}+x^*_{n-1}x_{n-1}$ is a unit for all $x_i$ of R; which answers a question raised in the literature recently.