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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.

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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