Skew Polynomial Rings and Regular Rings

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We considered a question raised by Armendariz, that is ; what can we say about a ring R and an injective module M when End_RM is simple Artinian? We proved that for an indecomposable injective right R-module M, if M is nonsingular, then End_RM is a division ring. We also proved the following : (1)If M is a completely injective indecomposable right ft-module of finite length, then EndRM is a division ring. (2)Let R be a semiprime right Goldie ring satisfying a polynomial identity andlet M be an indecompsable injective right R-module. Then End_RM is a division ring if and only if M is torsion-free. (3)Let ft be a commutative ring and let M be an indecomposable injective ft-module. Then End_RM is a division ring if and only if P = Ann_R (M) is a minimal prime ideal of R, Rp is a field and M * Rp. We characterized the ring with the property that the endomorphism ring of any indecomposable injective right R-module is a division ring. In particular, we proved that a commutative ring has this … More property if and only if R is von Neumann regularWe introduced a notion of the module of differentials of a noncommutative ring extension R/S and investigated their properties. We applied those to the theory of biderivations on semiprime rings and got some characterizations of symmetric biderivations on semiprime rings. of Let alpha be an automorphism of a ring ft and let 6 be an a-derivation of ft. A ring ftis strongly invariant in a skew polynomial ring R[CHI ; alpha, delta] if for any isomorphism psi of R[CHI ; alpha, delta] to another skew polynomial ring S[UPSILON, beta, **], there holds psi (R) = S.A ring ft is reduced if ft contains no nonzero nilpotent elements. A reduced ring ft with an automorphism a is a-reduced if, for any gamma epsilon R, gammaalpha(gamma) = 0 implies gamma= 0. We proved the following : Let ft be a strongly regular ring, let alpha be an automorphism of R, and let delta be an alpha-derivation of ft. Then ft is strongly invariant in R[CHI ; alpha, delta] if and only if ft is alpha-reduced. Less