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GENERALISED ARMENDARIZ PROPERTIES OF CROSSED PRODUCT TYPE

Part of: Rings and algebras arising under various constructions Conditions on elements Radicals and radical properties of rings

Published online by Cambridge University Press:  21 July 2015

LIANG ZHAO  and
YIQIANG ZHOU
LIANG ZHAO
Affiliation:
School of Mathematics and Physics, Anhui University of Technology, Maanshan 243032, P. R. China e-mail: lzhao78@gmail.com
YIQIANG ZHOU
Affiliation:
Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John's NFLD A1C 5S7, Canada e-mail: zhou@mun.ca
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Abstract

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Let R be a ring and M a monoid with twisting f:M × M → U(R) and action ω: M→ Aut(R). We introduce and study the concepts of CM-Armendariz and CM-quasi-Armendariz rings to generalise various Armendariz and quasi-Armendariz properties of rings by working on the context of the crossed product R*M over R. The following results are proved: (1) If M is a u.p.-monoid, then any M-rigid ring R is CM-Armendariz; (2) if I is a reduced ideal of an M-compatible ring R with M a strictly totally ordered monoid, then R/I being CM-Armendariz implies that R is CM-Armendariz; (3) if M is a u.p.-monoid and R is a semiprime ring, then R is CM-quasi-Armendariz. These results generalise and unify many known results on this subject.

MSC classification

Primary: 16S36: Ordinary and skew polynomial rings and semigroup rings
Secondary: 16N60: Prime and semiprime rings 16U99: None of the above, but in this section
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 58 , Issue 2 , May 2016 , pp. 313 - 323
Copyright
Copyright © Glasgow Mathematical Journal Trust 2015 

References

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