Modules over Noethrian rings and Abelian Groups

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(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.