Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
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A QUESTION ON ⁎-REGULAR RINGS

  • Cui, Jian (Department of Mathematics Anhui Normal University) ;
  • Yin, Xiaobin (Department of Mathematics Anhui Normal University)
  • Received : 2017.08.02
  • Accepted : 2018.07.06
  • Published : 2018.09.30
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Abstract

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.

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References

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