### A Cancellation Theorem for Artinian Local Algebras.

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Let $R$ be a commutative Noetherian ring and $\U0001d51e$ an ideal of $R$. We introduce the concept of $\U0001d51e$-weakly Laskerian $R$-modules, and we show that if $M$ is an $\U0001d51e$-weakly Laskerian $R$-module and $s$ is a non-negative integer such that ${\mathrm{Ext}}_{R}^{j}(R/\U0001d51e,{H}_{\U0001d51e}^{i}\left(M\right))$ is $\U0001d51e$-weakly Laskerian for all $i<s$ and all $j$, then for any $\U0001d51e$-weakly Laskerian submodule $X$ of ${H}_{\U0001d51e}^{s}\left(M\right)$, the $R$-module ${\mathrm{Hom}}_{R}(R/\U0001d51e,{H}_{\U0001d51e}^{s}\left(M\right)/X)$ is $\U0001d51e$-weakly Laskerian. In particular, the set of associated primes of ${H}_{\U0001d51e}^{s}\left(M\right)/X$ is finite. As a consequence, it follows that if $M$ is a finitely generated $R$-module and $N$ is an $\U0001d51e$-weakly...

The aim of this note is to give an alternative proof of uniqueness for the decomposition of a finitely generated torsion module over a P.I.D. (= principal ideal domain) as a direct sum of indecomposable submodules.Our proof tries to mimic as far as we can the standard procedures used when dealing with vector spaces.For the sake of completeness we also include a proof of the existence theorem.

Let R be a subring of the rationals. We want to investigate self splitting R-modules G, that is, such that $Ex{t}_{R}(G,G)=0$. For simplicity we will call such modules splitters (see [10]). Also other names like stones are used (see a dictionary in Ringel’s paper [8]). Our investigation continues [5]. In [5] we answered an open problem by constructing a large class of splitters. Classical splitters are free modules and torsion-free, algebraically compact ones. In [5] we concentrated on splitters which are larger...

Let R be a subring of the rational numbers ℚ. We recall from [3] that an R-module G is a splitter if $Ext{\xb9}_{R}(G,G)=0$. In this note we correct the statement of Main Theorem 1.5 in [3] and discuss the existence of non-free splitters of cardinality ℵ₁ under the negation of the special continuum hypothesis CH.

A closed loop parametrical identification procedure for continuous-time constant linear systems is introduced. This approach which exhibits good robustness properties with respect to a large variety of additive perturbations is based on the following mathematical tools: (1) module theory; (2) differential algebra; (3) operational calculus. Several concrete case-studies with computer simulations demonstrate the efficiency of our on-line identification scheme.