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• W4290805719 endingPage "474" @default.
• W4290805719 startingPage "464" @default.
• W4290805719 abstract "Alghazzawi and Leroy studied the structure of subsets satisfying the properties of symmetric and commutatively closed, that is, abc∈S for a,b,c∈R implies acb∈S, and ab∈S for a,b∈R implies ba∈S, respectively, where S is a subset of a ring R. In this article we discuss the structure of rings which are symmetric on zero (resp., idempotents). Such rings are also called symmetric (resp., I-symmetric). We first prove that if a polynomial ∑i=0maixi over a symmetric ring is a unit then a0 is a unit and ai is nilpotent for all i≥1; based on this result, we obtain that for a reduced ring R, the group of all units of the polynomial ring over R coincides with one of R, and that polynomial rings over I-symmetric rings are identity-symmetric. It is proved that for an abelian semiperfect ring R, R is I-symmetric if and only if the units in R form an Abelian group if and only if R is commutative. It is also proved that for an I-symmetric ring R, R is π-regular if and only if R/J(R) is a commutative regular ring and J(R) is nil, where J(R) is the Jacobson radical of R." @default.
• W4290805719 created "2022-08-12" @default.
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• W4290805719 date "2022-08-11" @default.
• W4290805719 modified "2023-09-29" @default.
• W4290805719 title "Symmetry on zero and idempotents" @default.
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• W4290805719 doi "https://doi.org/10.1080/00927872.2022.2102177" @default.
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