Reversible and pseudo-reversible rings
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This article concerns the structure of idempotents in reversi\-ble and pseudo-reversible rings in relation with various sorts of ring extensions. It is known that a ring $R$ is reversible if and only if $ab\in I(R)$ for $a, b\in R$ implies $ab=ba$; and a ring $R$ shall be said to be {\it pseudo-reversible} if $0\neq ab\in I(R)$ for $a, b\in R$ implies $ab=ba$, where $I(R)$ is the set of all idempotents in $R$. Pseudo-reversible is seated between reversible and quasi-reversible. It is proved that the reversibility, pseudo-reversibility, and quasi-reversibility are equivalent in Dorroh extensions and direct products. Dorroh extensions are also used to construct several sorts of rings which are necessary in the process.
Abstract
This article concerns the structure of idempotents in reversi\-ble and pseudo-reversible rings in relation with various sorts of ring extensions. It is known that a ring $R$ is reversible if and only if $ab\in I(R)$ for $a, b\in R$ implies $ab=ba$; and a ring $R$ shall be said to be {\it pseudo-reversible} if $0\neq ab\in I(R)$ for $a, b\in R$ implies $ab=ba$, where $I(R)$ is the set of all idempotents in $R$. Pseudo-reversible is seated between reversible and quasi-reversible. It is proved that the reversibility, pseudo-reversibility, and quasi-reversibility are equivalent in Dorroh extensions and direct products. Dorroh extensions are also used to construct several sorts of rings which are necessary in the process.
참고문헌 (14)
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