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GENERALIZED SEMI COMMUTATIVE RINGS AND THEIR EXTENSIONS

  • Published : 2008.05.31

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

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  3. Zero commutativity of nilpotent elements skewed by ring endomorphisms vol.45, pp.11, 2017, https://doi.org/10.1080/00927872.2017.1287267
  4. Zero Divisors in Skew Power Series Rings vol.43, pp.10, 2015, https://doi.org/10.1080/00927872.2014.946607
  5. STRUCTURE OF ZERO-DIVISORS IN SKEW POWER SERIES RINGS vol.52, pp.4, 2015, https://doi.org/10.4134/JKMS.2015.52.4.663
  6. QUASI-ARMENDARIZ PROPERTY FOR SKEW POLYNOMIAL RINGS vol.26, pp.4, 2011, https://doi.org/10.4134/CKMS.2011.26.4.557
  7. INSERTION-OF-FACTORS-PROPERTY ON SKEW POLYNOMIAL RINGS vol.52, pp.6, 2015, https://doi.org/10.4134/JKMS.2015.52.6.1161