Armendariz Properties Relative to a Ring Endomorphism

Abstract

For an endomorphism α of a ring R, we introduce the notion of an α-Armendariz ring to investigate the relative Armendariz properties. This concept extends the class of Armendariz rings and gives us an opportunity to study Armendariz rings in a general setting. It is obvious that every Armendariz ring is an α-Armendariz ring, but we shall give an example to show that there exists a right α-Armendariz ring which is not Armendariz. A number of properties of this version are established. It is shown that if I is a reduced ideal of a ring R such that R/I is a right α-Armendariz ring, then R is right α-Armendariz. For an endomorphism α of a ring R, we show that R is right α-Armendariz if and only if R[x] is right α-Armendariz. Moreover, a weak form of α-Armendariz rings is considered in the last section. We show that in general weak α-Armendariz rings need not be α-Armendariz.

Key Words:

• Armendariz rings
• α-Armendariz rings
• McCoy rings
• Weak α-Armendariz rings

2010 Mathematics Subject Classification:

• 13B02
• 16W20
• 16S50

ACKNOWLEDGMENTS

The authors would like to thank the referees for their careful reading of the article. This work was supported by the Provincial Natural Science Research Program of Higher Education Institution of Anhui Province of China (No. KJ2012Z028).

Notes

Communicated by A. Olshanskii.