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A generalization of symmetric ring property. (English) Zbl 1349.16061
Recall that a ring \(R\) is symmetric if for all \(r,s,t\in R\), \(rst=0\) implies \(rts=0\). In this paper, the authors restrict the choice of the variables to the subset \(N(R)\) of all nilpotent elements of \(R\) and investigate the consequences of doing so. In particular, \(R\) is called weak right nil-symmetric if for all \(a,b,c\in N(R)\), \(abc=0\) implies \(acb=0\). Weak left nil-symmetric is defined analogously and if \(R\) fulfills both these requirements, then it is called weak nil-symmetric. It is shown that in general, both these two requirements are necessary for a ring to be weak nil-symmetric. If a ring is weak right (or left) nil-symmetric, then its prime radical and nil radical coincide.
The authors discuss the relationships between these new notions and right (left) nil-symmetry and nil-semicommutativity. They also look at the behavior of weak left nil-symmetry and various ring extensions like matrix and polynomial rings. Many illuminating examples are given to illustrate some of the salient properties of these notions as well as its limitations.

MSC:
16U80 Generalizations of commutativity (associative rings and algebras)
16S70 Extensions of associative rings by ideals
16N40 Nil and nilpotent radicals, sets, ideals, associative rings
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