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In this paper, we introduce a new kind of rings that behave like semicommutative rings, but satisfy yet more known results. This kind of rings is called -semicommutative. We prove that a ring is -semicommutative if and only if is -semicommutative if and only if is -semicommutative. Also, if is -semicommutative, then is -semicommutative. The converse holds provided that is nilpotent and is power serieswise Armendariz. For each positive integer , is -semicommutative if and...
Let G be an abelian group and ◻ G its square subgroup as defined in the introduction. We show that the square subgroup of a non-homogeneous and indecomposable torsion-free group G of rank two is a pure subgroup of G and that G/◻ G is a nil group.
A -ring is strongly 2-nil--clean if every element in is the sum of two projections and a nilpotent that commute. Fundamental properties of such -rings are obtained. We prove that a -ring is strongly 2-nil--clean if and only if for all , is strongly nil--clean, if and only if for any there exists a -tripotent such that is nilpotent and , if and only if is a strongly -clean SN ring, if and only if is abelian, is nil and is -tripotent. Furthermore, we explore the structure...
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