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Keywords:
periodic element; semiclean element; 2-primal; polynomial ring
periodic element; semiclean element; 2-primal; polynomial ring
Summary:
We prove uniquely semiclean rings are reduced and hence Abelian. The semiclean elements of $R$, $R[x]$, and $M_n(R)$ 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 $R$.
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