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On nil-semicommutative rings. (English) Zbl 1253.16024

Summary: Semicommutative and Armendariz rings are a generalization of reduced rings, and therefore, nilpotent elements play an important role in this class of rings. There are many examples of rings with nilpotent elements which are semicommutative or Armendariz. In fact, [in Commun. Algebra 26, No. 7, 2265-2272 (1998; Zbl 0915.13001)], D. D. Anderson and V. Camillo prove that if \(R\) is a ring and \(n\geq 2\), then \(R[x]/(x^n)\) is Armendariz if and only if \(R\) is reduced.
In order to give a noncommutative generalization of the results of Anderson and Camillo, we introduce the notion of nil-semicommutative rings which is a generalization of semicommutative rings. If \(R\) is a nil-semicommutative ring, then we prove that \(ni\ell(R[x])=ni\ell(R)[x]\). It is also shown that nil-semicommutative rings are 2-primal, and when \(R\) is a nil-semicommutative ring, then the polynomial ring \(R[x]\) over \(R\) and the rings \(R[x]/(x^n)\) are weak Armendariz, for each positive integer \(n\), generalizing related results of Z.-K. Liu and R.-Y. Zhao, [Commun. Algebra 34, No. 7, 2607-2616 (2006; Zbl 1110.16026)].

MSC:

16S36 Ordinary and skew polynomial rings and semigroup rings
16N40 Nil and nilpotent radicals, sets, ideals, associative rings
16U80 Generalizations of commutativity (associative rings and algebras)
16W20 Automorphisms and endomorphisms
16E50 von Neumann regular rings and generalizations (associative algebraic aspects)
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