Generalized semicommutative rings and their extensions
2
조회
0
다운로드
5
인용
0
참고문헌
키워드
초록
For an endomorphism α of a ring R, the endomorphism α is called semicommutative if ab = 0 implies aRα (b) = 0 for a ∈ R. A ring R is called α-semicommutative if there exists a semicommutative endomor-phism α of R. In this paper, various results of semicommutative ringsare extended to α-semicommutative rings. In addition, we introduce thenotion of an α-skew power series Armendariz ring which is an extensionof Armendariz property in a ring R by considering the polynomials in theskew power series ring R[[x;α]]. We show that a number of interestingproperties of a ringR transfer to its the skew power series ringR[[x;α]]and vice-versa such as the Baer property and the p.p.-property, when R is α-skew power series Armendariz. Several known results relating to α-rigid rings can be obtained as corollaries of our results.
Abstract
For an endomorphism α of a ring R, the endomorphism α is called semicommutative if ab = 0 implies aRα (b) = 0 for a ∈ R. A ring R is called α-semicommutative if there exists a semicommutative endomor-phism α of R. In this paper, various results of semicommutative ringsare extended to α-semicommutative rings. In addition, we introduce thenotion of an α-skew power series Armendariz ring which is an extensionof Armendariz property in a ring R by considering the polynomials in theskew power series ring R[[x;α]]. We show that a number of interestingproperties of a ringR transfer to its the skew power series ringR[[x;α]]and vice-versa such as the Baer property and the p.p.-property, when R is α-skew power series Armendariz. Several known results relating to α-rigid rings can be obtained as corollaries of our results.
참고문헌 (15)
- D. D. Anderson, "Semigroups and rings whose zero products commute", Communications in Algebra, 27(6), pp.2847~2852 (1999)
- G. F. Birkenmeier, "Principally quasi-Baer rings", Communications in Algebra, 29(2), pp.639~660 (2001)
- W. E. Clark, "Twisted matrix units semigroup algebras", Duke Mathematical Journal, 34, pp.417~423 (1967)
- P. M. Cohn, "Reversible rings", Bulletin of the London Mathematical Society, 31(6), pp.641~648 (1999)
- J. M. Habeb, "A note on zero commutative and duo rings", Mathematical Journal of Okayama University, 32, pp.73~76 (1990)
- C. Y. Hong, "Ore extensions of Baer and p.p.-rings", Journal of Pure and Applied Algebra, 151(3), pp.215~226 (2000)
- C. Y. Hong, "On skew Armendariz rings", Communications in Algebra, 31(1), pp.103~122 (2003)
- C. Y. Hong, "Extensions of generalized reduced rings", Algebra Colloquium, 12(2), pp.229~240 (2005)
- C. Y. Hong, "Rigid ideals and radicals of Ore extensions", Algebra Colloquium, 12(3), pp.399~412 (2005)
- I. Kaplansky, "Rings of Operators" (1968), W. A. Benjamin, Inc.
- N. K. Kim, "Extensions of reversible rings", Journal of Pure and Applied Algebra, 185(1), pp.207~223 (2003)
- J. Krempa, "Some examples of reduced rings", Algebra Colloquium, 30(4), pp.289~300 (1996)
- J. Lambek, "On the representation of modules by sheaves of factor modules", Canadian Mathematical Bulletin, 14, pp.359~368 (1971)
- M. B. Rege, "Armendariz rings", Proceedings of the Japan Academy Series A Mathematical Sciences, 73(1), pp.14~17 (1997)
- G. Y. Shin, "Prime ideals and sheaf representation of a pseudo symmetric ring", Transactions of the American Mathematical Society, 184, pp.43~60 (1973)