Armendariz Properties Relative to a Ring Endomorphism |
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Authors: | Liang Zhao Xiaoguang Yan |
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Affiliation: | 1. School of Mathematics and Physics , Anhui University of Technology , Maanshan , China lzhao78@gmail.com;3. School of Mathematics &4. Information Technology , Nanjing Xiaozhuang University , Nanjing , China |
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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. |
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Keywords: | Armendariz rings α-Armendariz rings McCoy rings Weak α-Armendariz rings |
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