SKEW POLYNOMIAL RINGS OVER SEMIPRIME RINGS

Title & Authors
SKEW POLYNOMIAL RINGS OVER SEMIPRIME RINGS
Hong, Chan-Yong; Kim, Nam-Kyun; Lee, Yang;

Abstract
Y. Hirano introduced the concept of a quasi-Armendariz ring which extends both Armendariz rings and semiprime rings. A ring R is called quasi-Armendariz if $\small{a_iRb_j}$ = 0 for each i, j whenever polynomials $\small{f(x)\;=\;\sum_{i=0}^ma_ix^i}$, $\small{g(x)\;=\;\sum_{j=0}^mb_jx^j\;{\in}\;R[x]}$ satisfy f(x)R[x]g(x) = 0. In this paper, we first extend the quasi-Armendariz property of semiprime rings to the skew polynomial rings, that is, we show that if R is a semiprime ring with an epimorphism $\small{\sigma}$, then f(x)R[x; $\small{\sigma}$]g(x) = 0 implies $\small{a_iR{\sigma}^{i+k}(b_j)=0}$ for any integer k $\small{\geq}$ 0 and i, j, where $\small{f(x)\;=\;\sum_{i=0}^ma_ix^i}$, $\small{g(x)\;=\;\sum_{j=0}^mb_jx^j\;{\in}\;R[x,\;{\sigma}]}$. Moreover, we extend this property to the skew monoid rings, the Ore extensions of several types, and skew power series ring, etc. Next we define $\small{\sigma}$-skew quasi-Armendariz rings for an endomorphism $\small{\sigma}$ of a ring R. Then we study several extensions of $\small{\sigma}$-skew quasi-Armendariz rings which extend known results for quasi-Armendariz rings and $\small{\sigma}$-skew Armendariz rings.
Keywords
semiprime ring;quasi-Armendariz ring;skew polynomial ring;
Language
English
Cited by
1.
QUASI-ARMENDARIZ PROPERTY FOR SKEW POLYNOMIAL RINGS,;;

대한수학회논문집, 2011. vol.26. 4, pp.557-573
2.
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대한수학회지, 2015. vol.52. 6, pp.1161-1178
1.
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2.
Annihilator Ideals of Noncommutative Ring Constructions, Communications in Algebra, 2016, 44, 1, 63
3.
INSERTION-OF-FACTORS-PROPERTY ON SKEW POLYNOMIAL RINGS, Journal of the Korean Mathematical Society, 2015, 52, 6, 1161
4.
Quasi-Armendariz generalized power series rings, Journal of Algebra and Its Applications, 2016, 15, 05, 1650086
5.
GENERALISED ARMENDARIZ PROPERTIES OF CROSSED PRODUCT TYPE, Glasgow Mathematical Journal, 2016, 58, 02, 313
6.
Special properties of the ring Sn(R), Journal of Algebra and Its Applications, 2016, 1750212
7.
The Some Properties of Skew Polynomial Rings, Advances in Pure Mathematics, 2016, 06, 07, 507
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