It is proved that EJ is injective if E is an injective module over a valuation ring R, for each prime ideal J 6= Z. Moreover, if E or Z is flat, then EZ is injective too. It follows that localizations of injective modules over h-local Prüfer domains are injective too. If S is a multiplicative subset of a noetherian ring R, it is well known that SE is injective for each injective R-module E. The...

We construct certain systems of elements in K2 of some CM elliptic curves. When the classnumber of the field of CM is 1, The image of this system under the regulator map forms an euler system in the sense of Rubin [Ru]. §1. Systems of K2 of CM elliptic curves. Let K be an imaginary quadratic field. Take a Hecke character φ of K of type (1,0) and let fφ be its conductor. Let H = K(fφ) be the ray...

let ( r,m ) be a noetherian local ring, a an ideal of r and m a finitely generated r- module. we investigate some properties of formal local cohomology modules with respect to a serre subcategory. we provide a common language to indicate some properties of formal local cohomology modules. let ( r,m ) be a noetherian local ring, a an ideal of r and m a finitely generated r- module. we investigat...

Let (S*, k(S2)) be a knot formed by spinning a polyhedral arc a. about the standard 2-sphere S" in the 3-sphere S3. Then the second homotopy group of S"—k(S2) as a Z^-module is isomorphic to each of the following: (1) The fundamental ideal modulo the left ideal generated by a—\, where a is the image in ■¡r1(Si—k(S2)) of a generator of (2) The first homology group of the kernel of ^(SP-hiS1))-* ...

A generalization of injective modules (noted GI-modules), distinct from p-injective modules, is introduced. Rings whose p-injective modules are GI are characterized. If M is a left GI-module, E = End(AM), then E/J(E) is von Neumann regular, where J(E) is the Jacobson radical of the ring E. A is semisimple Artinian if, and only if, every left A-module is GI. If A is a left p. p., left GI-ring su...

Given free modules $M\subseteq L$ of finite rank $f\geq 1$ over a principal ideal domain $R$, we give procedure to construct basis $L$ from $M$ assuming the invariant factors or elementary divisors $L/M$ are known. matrix $A\in M_{m,n}(R)$ $r$, its nullspace in $R^n$ is $R$-module $f=n-r$. We submodule $f$ naturally associated with $A$ and whose easily computable, determine quotient module then...

The purpose of this paper is to study codes over finite principal ideal rings. To do this, we begin with codes over finite chain rings as a natural generalization of codes over Galois rings GR(pe, l) (including Zpe). We give sufficient conditions on the existence of MDS codes over finite chain rings and on the existence of self-dual codes over finite chain rings. We also construct MDS self-dual...

In this paper, we give an important isomorphism between contacyclic codes and cyclic codes,over finite principal ideal rings.Necessary and sufficient conditions for the existence of non-trivial cyclic self-dual codes over finite principal ideal rings are given.

It is shown that a commutative reduced ring R is a Baer ring if and only if it is a CS-ring; if and only if every dense subset of Spec (R) containing Max (R) is an extremally disconnected space; if and only if every non-zero ideal of R is essential in a principal ideal generated by an idempotent.