Modules over Noethrian rings and Abelian Groups

(1)众所周知，特征为0的域上的Weyl代数R是Noetherian简单代数，而不是Artinian。根据这个性质，我们知道在Weyl代数R上有限长度的模是循环的。从这个事实出发，我们考虑了一类具有这个性质的环。证明了环R的下列条件是等价的：R的每一个非零因子环都不是Artinian的；每一个有限长度的右r模都是循环的；每一个有限长度的左r模都是循环的；存在一个整数n，使得每一个有限长度的右r模都由n个元素生成；一个简单右r模的有限多个拷贝的直接和是循环的。我们称环R为flc环，如果每个长度有限的右R模都是循环的。证明了如果R是一个flc环，那么R的每一个有限的正则化扩展也是一个flc环。证明了FLC环类是森田稳定的。(2)设d为多项式环K[x_1，…]的K阶导数，x_n]在特征为0的域K上，设D为D对分数域K（x_1，…，x_n）的扩展。最近m.a ayad和P.Ryckelynck（2002）证明了：如果d的核函数Ker(d)包含n-1个代数无关多项式，则Ker(d)等于$Ker(d)的分数域Q（Ker(d)）。我们对这个结果做了一个简短的证明。(3)研究了含有有限生成p -内射左理想的环。我们证明了如果R包含一个有限生成的p -内射左理想I，使得R/I是完全可约的，并且如果R的每一个左半中心幂等都是中心的，那么R就是一个左p -内射环。作为这一结果的副产品，我们给出了具有非零环环的冯诺依曼正则环的一个新的表征。同时，我们也得到了半素数左noether环是Artinian的一个充分必要条件。

(1)It is well-known that a Weyl algebra R over a field of characteristic zero is a Noetherian simple algebra, but is not Artinian.. From this property, we know that a module of finite length over the Weyl algebra R is cyclic. From this fact, we considered the class of rings with this property. We proved that the following conditions are equivalent for a ring R : Every nonzero factor ring of R is not Artinian ; Every right R-module of finite length is cyclic ; Every left R-module of finite length is cyclic ; There exists an integer n such that every right R-module of finite length is generated by n-elements ; Every direct sum of finitely many copies of a simple right R-module is cyclic. We call a ring R a FLC-ring if every right R-module of finite length is cyclic. We proved that if R is a FLC-ring, then every finite normalizing extension of $R$ is also a FLC-ring. We also proved that the class of FLC rings is Morita stable.(2)Let d be a K-derivation of the polynomial ring K[x_1,...,x_n] over a field K of characteristic 0 and let D be the extension of d to the fraction field K(x_1,...,x_n). Recently M.Ayad and P.Ryckelynck (2002) proved the following : If the kernel Ker(d) of d contains n-1 algebraically independent polynomials, then Ker(D) is equal to the fraction field Q(Ker(D)) of $Ker(D). We gave a short proof for this result.(3)We studied for rings containing a finitely generated P-injective left ideal. We proved that if R contains a finitely generated P-injective left ideal I such that R/I is completely reducible, and if every left semicentral idempotent of R is central, then R is a left P-injective ring. As a byproduct of this result we gave a new characterization of a von Neumann regular ring with nonzero socle. Also we were able to find a necessary and sufficient condition for semiprime left Noetherian rings to be Artinian.