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NI rings and related properties. (English) Zbl 1342.16015
A ring \(R\) has been called an NI-ring if its nilradical \(N^*(R)=\sum(\text{nil ideals})\) coincides with the set \(N(R)\) of all its nilpotent elements. Such a ring clearly fulfills \(aRb\cup\{ab\}\subseteq N(R)\) for all \(a,b\in N(R)\), but, as it is shown in this paper, the converse does not necessarily hold. Consequently the authors call rings that do satisfy this property weak NI and the study of such rings is initiated in this paper.
The first results show that the class of weak NI rings is closed under subrings, direct limits with monomorphisms, direct products and direct sums, but not necessarily under homomorphic images and ideal extensions. In the next section, the relationships between weak NI rings and related extensions are investigated. In particular it is seen that extensions of this notion are fairly well-behaved. For example, if \(R\) is a weak NI ring, then so are several of the common subrings of matrix rings like the upper triangular matrix rings, the upper triangular matrix rings with common diagonal element, the circulants \(R[x]/\langle x^n\rangle\) as well as the Morita ring \(\Bigl[\begin{smallmatrix} R&M\\ 0&S\end{smallmatrix}\Bigr]\) with \(R\) and \(S\) weak NI. Moreover, if \(R\) is an Armendariz ring, then \(R\) weak NI implies that \(R[x]\) is also weak NI.
16N40 Nil and nilpotent radicals, sets, ideals, associative rings
16U80 Generalizations of commutativity (associative rings and algebras)
16S36 Ordinary and skew polynomial rings and semigroup rings
16S50 Endomorphism rings; matrix rings
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