WEAK α-SKEW ARMENDARIZ RINGS
2
조회
0
다운로드
0
인용
0
참고문헌
키워드
초록
For an endomorphism α of a ring R, we introduce the weak α-skew Armendariz rings which are a generalization of the α-skew Armendariz rings and the weak Armendariz rings, and investigate their properties. Moreover, we prove that a ring R is weak α-skew Armendariz if and only if for any n, the n × n upper triangular matrix ring Tn(R) is weak ¯α-skew Armendariz, where ¯α : Tn(R)→ Tn(R) is an extension of α. If R is reversible and α satisfies the condition that ab = 0 implies aα(b)=0 for any a, b ∈ R, then the ring R[x]/(xn) is weak ¯α-skew Armendariz,where (xn) is an ideal generated by xn, n is a positive integer and ¯α : R[x]/(xn)→ R[x]/(xn) is an extension of α. If α also satisfies the condition that αt = 1 for some positive integer t, the ring R[x] (resp,R[x; α]) is weak ¯α-skew (resp, weak) Armendariz, where ¯α : R[x] → R[x]is an extension of α.
Abstract
For an endomorphism α of a ring R, we introduce the weak α-skew Armendariz rings which are a generalization of the α-skew Armendariz rings and the weak Armendariz rings, and investigate their properties. Moreover, we prove that a ring R is weak α-skew Armendariz if and only if for any n, the n × n upper triangular matrix ring Tn(R) is weak ¯α-skew Armendariz, where ¯α : Tn(R)→ Tn(R) is an extension of α. If R is reversible and α satisfies the condition that ab = 0 implies aα(b)=0 for any a, b ∈ R, then the ring R[x]/(xn) is weak ¯α-skew Armendariz,where (xn) is an ideal generated by xn, n is a positive integer and ¯α : R[x]/(xn)→ R[x]/(xn) is an extension of α. If α also satisfies the condition that αt = 1 for some positive integer t, the ring R[x] (resp,R[x; α]) is weak ¯α-skew (resp, weak) Armendariz, where ¯α : R[x] → R[x]is an extension of α.
참고문헌 (9)
- D. D. Anderson, "Armendariz rings and Gaussian rings", Comm. Algebra, 26(7), pp.2265~2272 (1998)
- E. P. Armendariz, "A note on extensions of Baer and P.P.-rings", J. Austral. Math. Soc., 18, pp.470~473 (1974)
- J. L. Chen, "Extensions of injectivity and coherent rings", Comm. Algebra, 34(1), pp.275~288 (2006)
- C. Y. Hong, "On skew Armendariz rings", Comm. Algebra, 31(1), pp.103~122 (2003)
- C. Huh, "Armendariz rings and semicommutative rings", Comm. Algebra, 30(2), pp.751~761 (2002)
- N. K. Kim, "Armendariz rings and reduced rings", J. Algebra, 223(2), pp.477~488 (2000)
- N. K. Kim, "Extensions of reversible rings", J. Pure Appl. Algebra, 185(1), pp.207~223 (2003)
- Z. K. Liu, "On weak Armendariz rings", Comm. Algebra, 34(7), pp.2607~2616 (2006)
- M. B. Rege, "Armendariz rings", Proc. Japan Acad. Ser. A Math. Sci., 73(1), pp.14~17 (1997)