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Near-commutativity and partial-periodicity conditions for rings. (English) Zbl 1080.16028

Let \(R\) be a ring; and let \(N(R)\), \(Z(R)\), and \(J(R)\) denote respectively the set of nilpotent elements, the center, the Jacobson radical of \(R\). Let \(P(R)\) be the set of potent elements of \(R\) – i.e. \(P(R)=\{x\in R\mid x^n=x\) for some integer \(n=n(x)>1\}\). For \(x,y\in R\), the commutator \(xy-yx\) is denoted by \([x,y]\); and the extended commutators \([x,y]_k\) are defined inductively, letting \([x,y]_1=[x,y]\) and \([x,y]_{k+1}=[[x,y]_k,y]\). A ring \(R\) is called an Engel ring if for each \(x,y\in R\), there exists a positive integer \(k\) such that \([x,y]_k=0\).
In the paper under review, a ring \(R\) is called a weakly COPE ring if for each \(x,y\in R\), either \([x,y]_k=0\) for some positive integer \(k\) or there exists \(n=n(x,y)>1\) such that \(x^n-y^n\in N(R)\cap Z(R)\). It is easily observed that the class of Engel rings is contained in the class of weakly COPE rings. In [Result. Math. 42, No. 1-2, 28-31 (2002; Zbl 1036.16027)] the first author studied commutativity in COPE rings \(R\), which are defined by the property that for each \(a,b\in R\), either \(ab=ba\) or there exists an integer \(n=n(a,b)>1\) for which \(a^n=b^n\). In [A. Rosin, A. Yaqub, Int. J. Math. Math. Sci. 2003, No. 33, 2097-2107 (2003; Zbl 1038.16023)] a ring \(R\) was defined to be subweakly periodic if each \(x \in R\setminus J(R)\) can be expressed as a sum of a nilpotent element and a potent element; and some commutativity theorems for subweakly periodic rings were presented. In the paper under review, for a subset \(S\) of a ring \(R\), the authors define \(R\) to be \(S\)-subweakly periodic if \(R\setminus S\subseteq P(R)+N(R)\). Thus \(R\) is subweakly periodic if \(R\) is \(J(R)\)-subweakly periodic.
In the paper under review, the authors provide the following results for weakly COPE rings and \(S\)-subweakly periodic rings.
Theorem 2.1. Let \(R\) be a COPE ring with \(N(R)\) commutative. If \(R\) contains a proper additive subgroup \(S\) such that \(R\) is \(S\)-subweakly periodic, then \(R\) is commutative. Theorem 2.3. Let \(R\) be a ring with 1 such that for each \(a,b\in R\), either \([a,b]=0\) or there exists \(n>1\) such that \(a^n-b^n\in N(R)\cap Z(R)\). Then \(R\) is commutative.
Theorem 3.1. If \(R\) is a weakly COPE ring and \(N(R)\) is commutative, then \(N(R)\) is an ideal and \(R/N(R)\) is commutative.
Theorem 3.4. Let \(R\) be a weakly COPE ring containing a commutative ideal \(I\) such that \(N(R)\subseteq I\) and \(R\) is \(I\)-subweakly periodic. Then \(R\) is commutative.
Theorem 4.1. Let \(R\) be a ring in which \(N(R)\subseteq Z(R)\). If \(N(R)\) is contained in a commutative ideal \(I\) for which \(R\) is \(I\)-subweakly periodic, then \(R\) is commutative.

MSC:

16U80 Generalizations of commutativity (associative rings and algebras)
16U70 Center, normalizer (invariant elements) (associative rings and algebras)
16N40 Nil and nilpotent radicals, sets, ideals, associative rings
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References:

[1] Bell, H.E.: On some commutativity theorems of Herstein. Arch. Math. 24 (1973), 34–38. · Zbl 0251.16021 · doi:10.1007/BF01228168
[2] Bell, H.E.: A near-commutativity property for rings. Result. Math. 42 (2002), 28–31. · Zbl 1036.16027 · doi:10.1007/BF03323550
[3] Chuang, C-L and Lin, J-S: On a conjecture by Herstein. J. Algebra 126 (1989), 119–138. · Zbl 0688.16036 · doi:10.1016/0021-8693(89)90322-0
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[6] Rosin, A. and Yaqub, A.: Weakly periodic and subweakly periodic rings. Internat. J. Math. & Math. Sci 33 (2003), 2097–2107. · Zbl 1038.16023 · doi:10.1155/S0161171203210589
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