Si considerano le estensioni chiuse $B$ di un $R$-modulo $A$ mediante un $R$-modulo $C$ nel caso in cui $R$ sia un anello semi-artiniano, cioè un anello $R$ con la proprietà che per ogni quoziente $(R{/}_I)\ne 0$ sia soc $(R/I)\ne 0$. Tali estensioni sono caratterizzate dal fatto che $A$ deve essere un sottomodulo semi-puro di $B$.

Let M be a finitely generated module over an Artin algebra. By considering the lengths of the modules in the minimal projective resolution of M, we obtain the Betti sequence of M. This sequence must be bounded if M is eventually periodic, but the converse fails to hold in general. We give conditions under which it holds, using techniques from Hochschild cohomology. We also provide a result which under certain conditions guarantees the existence of periodic modules. Finally, we study the case when...

Let $C_F\left(X\right)$ be the socle of C(X). It is shown that each prime ideal in $C\left(X\right)/C_F\left(X\right)$ is essential. For each h ∈ C(X), we prove that every prime ideal (resp. z-ideal) of C(X)/(h) is essential if and only if the set Z(h) of zeros of h contains no isolated points (resp. int Z(h) = ∅). It is proved that $dim(C\left(X\right)/C_F\left(X\right))\ge dimC\left(X\right)$, where dim C(X) denotes the Goldie dimension of C(X), and the inequality may be strict. We also give an algebraic characterization of compact spaces with at most a countable number of nonisolated points. For each essential...