SKEW POLYNOMIAL RINGS OVER SEMIPRIME RINGS

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 a(i)Rb(j) = 0 for each i, j whenever polynomials f(x) = Sigma(m)(i=0) a(i)x(i), g(x) = Sigma(n)(j=0) b(j)x(j) is an element of 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 sigma, then f(x)R[x; sigma]g(x) = 0 implies a(i)R sigma(i+k) (b(j)) = 0 for any integer k >= 0 and i, j, where f(x) = Sigma(m)(i=0)a(i)x(i), g(x) = Sigma(n)(j=0)b(j)x(j) is an element of 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 sigma-skew quasi-Armendariz rings for an endomorphism sigma of a ring R. Then we study several extensions of sigma-skew quasi-Armendariz rings which extend known results for quasi-Armendariz rings and sigma-skew Armendariz rings.