We consider connections between similar sublattices and coincidence site lattices and, more generally, between similar submodules and coincidence site modules of general (free) Z-modules in R. In particular, we generalise results obtained by S. Glied and M. Baake on similarity and coincidence isometries of lattices and certain lattice-like modules called S-modules. An important result is that t...

let $r=oplus_{nin bbb n_0}r_n$ be a noetherian homogeneous ring with local base ring $(r_0,frak{m}_0)$, $m$ and $n$ two finitely generated graded $r$-modules. let $t$ be the least integer such that $h^t_{r_+}(m,n)$ is not minimax. we prove that $h^j_{frak{m}_0r}(h^t_{r_+}(m,n))$ is artinian for $j=0,1$. also, we show that if ${rm cd}(r_{+},m,n)=2$ and $tin bbb n_0$, then $h^t_{frak{m}_0r}(h^2_{...

All rings are commutative with identity and all modules are unitary. In this note we give some properties of a finite collection of submodules such that the sum of any two distinct members is multiplication, generalizing those which characterize arithmetical rings. Using these properties we are able to give a concise proof of Patrick Smith’s theorem stating conditions ensuring that the sum and ...

The main aim of this paper is to show that an AB5 module whose small submodules have Krull dimension has a radical having Krull dimension. The proof uses the notion of dual Goldie dimension.

We give a characterization of Σ-cotorsion modules over valuation domains in terms of descending chain conditions on certain chains of definable subgroups. We prove that pure submodules, direct products and modules elementarily equivalent to a Σ-cotorsion module are again Σ-cotorsion. Moreover, we describe the structure of Σ-cotorsion modules.

Let R be a commutative Noetherian ring of prime characteristic p. Assume R (or, more generally, a finitely generated R-module N with SuppR(N) = Spec(R)) has finite F-representation type (abbreviated FFRT) by finitely generated R-modules. Then, for every c ∈ R◦, there is a uniform test exponent Q = p for c and for all R-modules. As a consequence, we show the existence of uniform test exponents o...