GENERALIZED SEMI COMMUTATIVE RINGS AND THEIR EXTENSIONS

Title & Authors
GENERALIZED SEMI COMMUTATIVE RINGS AND THEIR EXTENSIONS
Baser, Muhittin; Harmanci, Abdullah; Kwak, Tai-Keun;

Abstract
For an endomorphism $\small{{\alpha}}$ of a ring R, the endomorphism $\small{{\alpha}}$ is called semicommutative if ab=0 implies $\small{aR{\alpha}(b)}$=0 for a $\small{{\in}}$ R. A ring R is called $\small{{\alpha}}$-semicommutative if there exists a semicommutative endomorphism $\small{{\alpha}}$ of R. In this paper, various results of semicommutative rings are extended to $\small{{\alpha}}$-semicommutative rings. In addition, we introduce the notion of an $\small{{\alpha}}$-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 $\small{R[[x;\;{\alpha}]]}$. We show that a number of interesting properties of a ring R transfer to its the skew power series ring $\small{R[[x;\;{\alpha}]]}$ and vice-versa such as the Baer property and the p.p.-property, when R is $\small{{\alpha}}$-skew power series Armendariz. Several known results relating to $\small{{\alpha}}$-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;
Language
English
Cited by
1.
ON QUASI-RIGID IDEALS AND RINGS,;;;

대한수학회보, 2010. vol.47. 2, pp.385-399
2.
QUASI-ARMENDARIZ PROPERTY FOR SKEW POLYNOMIAL RINGS,;;

대한수학회논문집, 2011. vol.26. 4, pp.557-573
3.
STRUCTURE OF ZERO-DIVISORS IN SKEW POWER SERIES RINGS,;;;

대한수학회지, 2015. vol.52. 4, pp.663-683
4.
INSERTION-OF-FACTORS-PROPERTY ON SKEW POLYNOMIAL RINGS,;;;;;

대한수학회지, 2015. vol.52. 6, pp.1161-1178
1.
ON QUASI-RIGID IDEALS AND RINGS, Bulletin of the Korean Mathematical Society, 2010, 47, 2, 385
2.
Generalized Semicommutative and Skew Armendariz Ideals, Ukrainian Mathematical Journal, 2015, 66, 9, 1354
3.
Zero commutativity of nilpotent elements skewed by ring endomorphisms, Communications in Algebra, 2017, 45, 11, 4881
4.
Zero Divisors in Skew Power Series Rings, Communications in Algebra, 2015, 43, 10, 4427
5.
STRUCTURE OF ZERO-DIVISORS IN SKEW POWER SERIES RINGS, Journal of the Korean Mathematical Society, 2015, 52, 4, 663
6.
QUASI-ARMENDARIZ PROPERTY FOR SKEW POLYNOMIAL RINGS, Communications of the Korean Mathematical Society, 2011, 26, 4, 557
7.
INSERTION-OF-FACTORS-PROPERTY ON SKEW POLYNOMIAL RINGS, Journal of the Korean Mathematical Society, 2015, 52, 6, 1161
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