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On conditions provided by nilradicals. (English) Zbl 1182.16015
A ring \(R\) is called IFP, if \(ab=0\) implies \(aRb=0\) for \(a,b\in R\). As the IFP condition is not preserved by polynomial ring extensions, in this paper the authors introduce and study a generalized condition of the IFPness that can be lifted up to polynomial rings. Namely, they call a ring \(R\) quasi-IFP if \(\sum_{i=0}^nRa_iR\) is nilpotent whenever \(\sum_{i=0}^na_ix^i\in R[x]\) is nilpotent, where \(R[x]\) denotes the polynomial ring over \(R\) in the indeterminate \(x\).
The authors show that IFP rings are quasi-IFP but this implication is irreversible. They give various conditions that are equivalent to the quasi-IFPness and also study conditions under which various concepts near to the quasi-IFPness coincide providing a wealth of examples. They show, for example, that if \(R\) is a semiprime ring then \(R\) is quasi-IFP if and only if \(R\) is reduced. The authors thoroughly examine the structure of quasi-IFP rings and describe quasi-IFP rings via minimal strongly prime ideals. They obtain a useful method by which given rings are examined to be quasi-IFP. They close the paper by describing the structure of noncommutative quasi-IFP rings having smallest cardinality.

MSC:
16N40 Nil and nilpotent radicals, sets, ideals, associative rings
16S36 Ordinary and skew polynomial rings and semigroup rings
16D25 Ideals in associative algebras
16P60 Chain conditions on annihilators and summands: Goldie-type conditions
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