Skew Polynomial Rings and Regular Rings

我们审议了阿曼达里兹提出的一个问题，即；当End_RM是单Artinian时，对于环R和内射模M我们能说些什么呢？证明了对于不可分解内射右r模M，如果M是非奇异的，则End_RM是一个除环。我们还证明了：(1)如果M是一个有限长度的完全内射不可分解右ft模，则EndRM是一个除环。(2)设R为满足多项式恒等式的半素数右Goldie环，M为不可分解的内射右R模。则End_RM是除环当且仅当M是无扭环。(3)设ft为交换环，M为不可分解内射ft-模。则End_RM是除环当且仅当P = Ann_R (M)是R的极小素理想，Rp是域且M * Rp。利用任意不可分解内射右r模的自同态环是除环的性质对环进行了刻画。特别地，我们证明了一个可交换环当且仅当R是von Neumann正则，我们引入了一个非可交换环扩展R/S的微分模的概念，并研究了它们的性质。将其应用于半素环上的双推理论，得到了半素环上对称双推的一些刻画。设α是环ft的自同构，设6是环ft的a阶导数。环f是斜多项式环R中的强不变环[CHI；，[]如果对于R[CHI]的任意同构；到另一个歪斜多项式环S[UPSILON， beta, **]，则存在psi (R) = sa。如果环ft不包含非零幂零元素，则环ft约简。一个具有自同构A的约简环ft是约简的，如果对于任意R，（） = 0意味着= 0。我们证明了：设ft是一个强正则环，设是R的自同构，设是ft的α导，则ft在R[CHI]中是强不变的；当且仅当ft是约化后的。少

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