# zbMATH — the first resource for mathematics

On $$\alpha$$-nilpotent elements and $$\alpha$$-Armendariz rings. (English) Zbl 1323.16018
A well-known result of Armendariz shows that a reduced ring $$R$$ has the property that if $$f(x)g(x)=0$$ for $$f(x),g(x)\in R[x]$$, then $$ab=0$$ for any coefficients $$a$$ and $$b$$ of $$f(x)$$ and $$g(x)$$, respectively. Rings with this latter property are now called Armendariz rings. Subsequently this notion, as well as the set of nilpotent elements in an Armendariz ring, have been generalized and extended in many ways; in particular also to skew polynomial rings.
In this paper the authors study and add to such generalizations. In particular they investigate $$\overline\alpha$$-nilpotent elements in a skew polynomial ring $$R[x;\alpha]$$, where $$\overline\alpha$$ is the monomorphism induced by the monomorphism $$\alpha$$ of an Armendariz ring $$R$$. Furthermore, for an endomorphism $$\beta$$ of $$R$$, the notion $$\beta$$-nil-Armendariz is defined as a natural generalization of a nil-Armendariz ring to a skew-polynomial ring $$R[x;\beta]$$. Many examples are provided and the main line of investigation is to see how this notion transfers to related rings.
##### MSC:
 16N40 Nil and nilpotent radicals, sets, ideals, associative rings 16S36 Ordinary and skew polynomial rings and semigroup rings 16P60 Chain conditions on annihilators and summands: Goldie-type conditions 16U80 Generalizations of commutativity (associative rings and algebras)
Full Text:
##### References:
 [1] DOI: 10.1080/00927879808826274 · Zbl 0915.13001 · doi:10.1080/00927879808826274 [2] DOI: 10.1080/00927879908826596 · Zbl 0929.16032 · doi:10.1080/00927879908826596 [3] DOI: 10.1016/j.jalgebra.2008.01.019 · Zbl 1157.16007 · doi:10.1016/j.jalgebra.2008.01.019 [4] DOI: 10.1017/S1446788700029190 · Zbl 0292.16009 · doi:10.1017/S1446788700029190 [5] DOI: 10.4134/BKMS.2011.48.1.157 · Zbl 1209.16018 · doi:10.4134/BKMS.2011.48.1.157 [6] DOI: 10.1016/S0022-4049(99)00020-1 · Zbl 0982.16021 · doi:10.1016/S0022-4049(99)00020-1 [7] DOI: 10.1081/AGB-120016752 · Zbl 1042.16014 · doi:10.1081/AGB-120016752 [8] DOI: 10.1142/S100538670600023X · Zbl 1095.16014 · doi:10.1142/S100538670600023X [9] DOI: 10.1080/00927877608822125 · Zbl 0328.16001 · doi:10.1080/00927877608822125 [10] DOI: 10.1006/jabr.1999.8017 · Zbl 0957.16018 · doi:10.1006/jabr.1999.8017 [11] Krempa J., Algebra Colloq. 3 pp 289– (1996) [12] DOI: 10.1080/00927879708826000 · Zbl 0879.16016 · doi:10.1080/00927879708826000 [13] DOI: 10.4153/CMB-1971-065-1 · Zbl 0217.34005 · doi:10.4153/CMB-1971-065-1 [14] DOI: 10.1081/AGB-120037221 · Zbl 1068.16037 · doi:10.1081/AGB-120037221 [15] McConnell J. C., Noncommutative Noetherian Rings (1987) · Zbl 0644.16008 [16] DOI: 10.3792/pjaa.73.14 · Zbl 0960.16038 · doi:10.3792/pjaa.73.14
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.