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Skew Polynomial Rings and Regular Rings

斜多项式环和正则环

基本信息

批准号：
09640039
负责人：
HIRANO Yasuyuki
金额：
$ 1.92万
依托单位：
Okayama University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
1997
资助国家：
日本
起止时间：
1997 至 1998
项目状态：
已结题

项目摘要

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

我们考虑了Armendariz提出的一个问题，即当End_RM是单Artin的时候，我们能对环R和内射模M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是一个除环。（1）若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是挠自由的。（3）设ft是交换环，M是不可分解的内射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.我们刻画了这类环的一个性质：任何不可分解的内射右R-模的自同态环是一个除环。特别地，我们证明了交换环有这个 ...更多信息引入了非交换环扩张R/S的微分模的概念，并研究了它们的性质。并将其应用到半质环上的双导子理论中，得到了半质环上对称双导子的一些刻画。设α是环FT的自同构，δ是FT的a-导子.环f在斜多项式环R [CHI; alpha，delta]中是强不变的，如果对R [CHI; alpha，delta]到另一个斜多项式环S [UPSILON，beta，**]的任何同构psi，有psi（R）= S.环f是约化的，如果f不含非零幂零元.一个具有自同构a的约化环ft是a-约化的，如果对任何gamma ∈ R，gammaalpha（gamma）= 0蕴涵gamma = 0。证明了：设ft是强正则环，α是R的自同构，δ是ft的α-导子。然后ft在R [CHI; alpha，delta]中是强不变的当且仅当ft是alpha-约化的。少