Power Series Rings Satisfying a Zero Divisor Property |
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Authors: | Nam Kyun Kim Ki Hwan Lee |
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Affiliation: | 1. College of Liberal Arts , Hanbat National University , Daejeon, South Korea;2. Department of Mathematics , Pusan National University , Pusan, South Korea |
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Abstract: | In this note we continue to study zero divisors in power series rings and polynomial rings over general noncommutative rings. We first construct Armendariz rings which are not power-serieswise Armendariz, and find various properties of (power-serieswise) Armendariz rings. We show that for a semiprime power-serieswise Armendariz (so reduced) ring R with a.c.c. on annihilator ideals, R[[x]] (the power series ring with an indeterminate x over R) has finitely many minimal prime ideals, say B 1,…,B m , such that B 1… B m = 0 and B i = A i [[x]] for some minimal prime ideal A i of R for all i, where A 1,…,A m are all minimal prime ideals of R. We also prove that the power-serieswise Armendarizness is preserved by the polynomial ring extension as the Armendarizness, and construct various types of (power-serieswise) Armendariz rings. |
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Keywords: | Polynomial ring Power series ring (Power-serieswise) Armendariz ring Reduced ring Zero divisor |
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