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<p>We prove uniquely semiclean rings are reduced and hence Abelian. The semi-clean elements of <i>R</i>, <i>R</i>[<i>x</i>], and <i>M</i><sub><i>n</i></sub>(<i>R</i>) are compared. In particular, we show that the indeterminate <i>x</i> is never semiclean. The 2-primality of a ring <i>R</i> is characterized via the semiclean elements of <i>R</i>[<i>x</i>]. We also consider the quasi-clean elements of a ring <i>R</i> and compare them with the semiclean elements of <i>R</i>.</p>

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Semiclean and quasi-clean elements in rings

  • 期刊论文

  • 发表日期：

    2025年11月18日

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  • Guanglin Ma,
  • André Leroy

摘要

We prove uniquely semiclean rings are reduced and hence Abelian. The semi-clean elements of R, R[x], and Mn(R) are compared. In particular, we show that the indeterminate x is never semiclean. The 2-primality of a ring R is characterized via the semiclean elements of R[x]. We also consider the quasi-clean elements of a ring R and compare them with the semiclean elements of R.