Skew Polynomial Rings and Regular Rings

我们考虑了Armendariz提出的一个问题，即：当end_Rm是单Artin时，关于环R和内射模M，我们能说什么？证明了对于不可分解的内射右R-模M，如果M是非奇异的，则End_Rm是除环。证明了：(1)如果M是有限长的完全内射不可分解右ft模，则EndRm是除环。(2)设R是满足多项式恒等式的半素右Goldie环，M是不可分解的内射右R-模。则End_Rm是除环当且仅当M是无挠的。(3)设ft是交换环，M是不可分解内射ft-模。则END_RM是除环当且仅当P=ANN_R(M)是R的极小素数理想，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刻画了环。特别地，我们证明了交换环有这个…更多性质当且仅当R是von Neumann正则我们引入了非对易环扩张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导子理论，得到了半素环上对称双导子的一些刻画。设α是环FT的自同构，6是FT的a-导子。一个环ff在一个斜多项式环R[ch；α，Delta]中是强不变的，如果对R[ch；α，Delta]到另一个斜多项式环S[Upsilon，beta，**]的任何同构psi，则存在psi(R)=S。如果环ft不包含非零幂零元，则环ft被约化。具有自同构a的约化环Ft是a-约化的，如果对任何GammaεR，Gammaα(Gamma)=0意味着Gamma=0。我们证明了：设f是强正则环，α是R的自同构环，Delta是ft的α导子。则ft在R[ch；α，Delta]中是强不变的当且仅当ft是α-约化的。较少

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