# GENERALIZED SEMI COMMUTATIVE RINGS AND THEIR EXTENSIONS

• Published : 2008.05.31
• 86 9

#### Abstract

For an endomorphism ${\alpha}$ of a ring R, the endomorphism ${\alpha}$ is called semicommutative if ab=0 implies $aR{\alpha}(b)$=0 for a ${\in}$ R. A ring R is called ${\alpha}$-semicommutative if there exists a semicommutative endomorphism ${\alpha}$ of R. In this paper, various results of semicommutative rings are extended to ${\alpha}$-semicommutative rings. In addition, we introduce the notion of an ${\alpha}$-skew power series Armendariz ring which is an extension of Armendariz property in a ring R by considering the polynomials in the skew power series ring $R[[x;\;{\alpha}]]$. We show that a number of interesting properties of a ring R transfer to its the skew power series ring $R[[x;\;{\alpha}]]$ and vice-versa such as the Baer property and the p.p.-property, when R is ${\alpha}$-skew power series Armendariz. Several known results relating to ${\alpha}$-rigid rings can be obtained as corollaries of our results.

#### Keywords

semicommutative rings;rigid rings;skew power series rings;extended Armendariz rings;Baer rings;p.p.-rings

#### References

1. D. D. Anderson and V. Camillo, Semigroups and rings whose zero products commute, Comm. Algebra 27 (1999), no. 6, 2847-2852 https://doi.org/10.1080/00927879908826596
2. G. F. Birkenmeier, J. Y. Kim, and J. K. Park, Principally quasi-Baer rings, Comm. Algebra 29 (2001), no. 2, 639-660 https://doi.org/10.1081/AGB-100001530
3. W. E. Clark, Twisted matrix units semigroup algebras, Duck Math. J. 34 (1967), 417-423
4. P. M. Cohn, Reversible rings, Bull. London Math. Soc. 31 (1999), no. 6, 641-648 https://doi.org/10.1112/S0024609399006116
5. J. M. Habeb, A note on zero commutative and duo rings, Math. J. Okayama Univ. 32 (1990), 73-76
6. C. Y. Hong, N. K. Kim, and T. K. Kwak, Ore extensions of Baer and p.p.-rings, J. Pure Appl. Algebra 151 (2000), no. 3, 215-226 https://doi.org/10.1016/S0022-4049(99)00020-1
7. C. Y. Hong, N. K. Kim, and T. K. Kwak, On skew Armendariz rings, Comm. Algebra 31 (2003), no. 1, 103-122 https://doi.org/10.1081/AGB-120016752
8. C. Y. Hong, N. K. Kim, and T. K. Kwak, Extensions of generalized reduced rings, Algebra Colloq. 12 (2005), no. 2, 229-240 https://doi.org/10.1142/S1005386705000222
9. C. Y. Hong, T. K. Kwak, and S. T. Rizvi, Rigid ideals and radicals of Ore extensions, Algebra Colloq. 12 (2005), no. 3, 399-412 https://doi.org/10.1142/S1005386705000374
10. I. Kaplansky, Rings of Operators, W. A. Benjamin, Inc., New York-Amsterdam, 1968
11. N. K. Kim and Y. Lee, Extensions of reversible rings, J. Pure Appl. Algebra 185 (2003), no. 1-3, 207-223 https://doi.org/10.1016/S0022-4049(03)00109-9
12. J. Krempa, Some examples of reduced rings, Algebra Colloq. 3 (1996), no. 4, 289-300
13. J. Lambek, On the representation of modules by sheaves of factor modules, Canad. Math. Bull. 14 (1971), 359-368 https://doi.org/10.4153/CMB-1971-065-1
14. G. Y. Shin, Prime ideals and sheaf representation of a pseudo symmetric ring, Trans. Amer. Math. Soc. 184 (1973), 43-60 https://doi.org/10.2307/1996398
15. M. B. Rege and S. Chhawchharia, Armendariz rings, Proc. Japan Acad. Ser. A Math. Sci. 73 (1997), no. 1, 14-17

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