Let $R$ be a non-maximal order in a finite algebraic number field with integral closure $\overline{R}$. Although $R$ is not a unique factorization domain, we obtain a positive integer $N$ and a family $𝒬$ (called a Cale basis) of primary irreducible elements of $R$ such that $x^N$ has a unique factorization into elements of $𝒬$ for each $x\in R$ coprime with the conductor of $R$. Moreover, this property holds for each nonzero $x\in R$ when the natural map $\mathrm{Spec}\left(\overline{R}\right)\to \mathrm{Spec}\left(R\right)$ is bijective. This last condition is actually equivalent to several properties linked...

Let R be a commutative multiplication ring and let N be a non-zero finitely generated multiplication R-module. We characterize certain prime submodules of N. Also, we show that N is Cohen-Macaulay whenever R is Noetherian.

Every relatively convex-compact convex subset of a locally convex space is contained in a Banach disc. Moreover, an upper bound for the class of sets which are contained in a Banach disc is presented. If the topological dual $E^{\text{'}}$ of a locally convex space $E$ is the $\sigma (E^{\text{'}},E)$-closure of the union of countably many $\sigma (E^{\text{'}},E)$-relatively countably compacts sets, then every weakly (relatively) convex-compact set is weakly (relatively) compact.