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Commutative weakly nil clean unital rings. (English) Zbl 1316.16028

In this short and interesting paper the authors, generalizing the notions of weakly clean ring and nil clean ring, define weakly nil clean rings and investigate their properties for commutative rings. They prove that every weakly nil clean ring is zero-dimensional and hence a clean ring. They prove that the class of all weakly nil clean rings is closed under taking homomorphic images but not closed under taking finite direct products. They prove that a weakly nil clean ring with 2 nilpotent is a clean ring. Their main theorem asserts that a ring \(R\) is weakly nil clean if and only if \(R/N(R)\) is isomorphic to either a Boolean ring or \(Z_3\) or a product of two such rings.
Next they study weakly nil clean group rings. They completely characterize weakly nil clean group rings. They prove that a group ring \(R[G]\) (for \(G\) abelian) is weakly nil clean if and only if exactly one of the following three conditions is satisfied. (i) \(R\) is nil clean and \(G\) is a nontrivial torsion 2-group; (ii) \(R/N(R)\) is isomorphic to \(Z_3\) and \(G\) is a nontrivial torsion 3-group; (iii) \(R\) is weakly nil clean and \(G\) is trivial.
Finally, the authors study extension rings \(R[X]\), \(R(X)\), \(R\langle X\rangle\) and \(R[[X]]\). They prove that none of these rings is weakly nil clean. The authors conclude the paper with some open problems.

MSC:

16U60 Units, groups of units (associative rings and algebras)
13A99 General commutative ring theory
16S34 Group rings
16N40 Nil and nilpotent radicals, sets, ideals, associative rings
16U80 Generalizations of commutativity (associative rings and algebras)
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