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 aiRbj = 0 for each i,j whenever polynomials f(x) = Pm=0 aix i ,g(x) = Pn=0 bjx j 2 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 ae, then f(x)R(x;ae)g(x) = 0 im- plies aiRae i+k (bj) = 0 for any integer k ‚ 0 and i,j, where f(x) = Pm i=0 aix i ,g(x) = Pn j=0 bjx j 2 R(x;ae). Moreover, we extend this prop- erty to the skew monoid rings, the Ore extensions of several types, and skew power series ring, etc. Next we define ae-skew quasi-Armendariz rings for an endomorphism ae of a ring R. Then we study several exten- sions of ae-skew quasi-Armendariz rings which extend known results for quasi-Armendariz rings and ae-skew Armendariz rings. Throughout this paper R denotes an associative ring with identity. We denote by R(x) the polynomial ring with an indeterminate x over R. Rege and Chhawchharia (18) introduced the notion of an Armendariz ring. A ring R is called Armendariz if whenever polynomials f(x) = Pm=0 aix i ,g(x) = Pn j=0 bjx j 2 R(x) satisfy f(x)g(x) = 0, then aibj = 0 for each i,j. The name "Armendariz ring" was chosen from the fact that Armendariz (2, Lemma 1) had showed that a reduced ring (i.e., a ring without nonzero nilpotent elements) sat- isfies this condition. Many properties of Armendariz rings have been studied by several authors (1, 8, 10, 11, 12). Hirano (5) introduced a quasi-Armendariz ring which is generalizing an Armendariz ring. A ring R is called quasi-Armendariz if whenever polynomials f(x) = P m=0 aix i ,g(x) = P n=0 bjx j 2 R(x) satisfy f(x)R(x)g(x) = 0, then aiRbj = 0 for each i,j. Hirano (5, Corollary 3.8) proved that semiprime rings are quasi-Armendariz rings. Moreover, he showed that the class of quasi-Armendariz rings is Morita stable (4, Theorem 3.12 and Proposition 3.13), and that if R is a quasi-Armendariz ring, then some exten- sions of R (e.g., the n-by-n upper triangular matrix ring, the polynomial ring) are also quasi-Armendariz rings. But most of these properties are not stable in Armendariz rings (for example, (10, Examples 1 and 3, etc.)).