The goal of this paper is to give characterizations of minimal primary decompositions and radicals of submodules of any finitely generated module over a PID. To do this, we first consider free modules of finite rank and we characterize prime, primary and maximal submodules as well as minimal primary decompositions and radicals.

Analytical formulas are established for calculating the maximal and minimal ranks of the matrix-valued function A+BXC when the rank of X is fixed. Some consequences are also given.

In this paper the family of elliptic curves over Q given by the equation Ep :Y2 = (X - p)3 + X3 + (X + p)3 where p is a prime number, is studied. Itis shown that the maximal rank of the elliptic curves is at most 3 and someconditions under which we have rank(Ep(Q)) = 0 or rank(Ep(Q)) = 1 orrank(Ep(Q))≥2 are given.

We classify joinings between a fairly general class of higher rank diagonalizable actions on locally homogeneous spaces. In particular, we classify joinings of the action of a maximal R-split torus on G/Γ, with G a simple Lie group of R-rank ≥ 2 and Γ < G a lattice. We deduce from this a classification of measurable factors of such actions, as well as certain equidistribution properties.

— In this article, we study the maximal cross number of long zero-sumfree sequences in a finite Abelian group. Regarding this inverse-type problem, we formulate a general conjecture and prove, among other results, that this conjecture holds true for finite cyclic groups, finite Abelian p-groups and for finite Abelian groups of rank two. Also, the results obtained here enable us to improve, via ...

— In this paper, we study the maximal cross number of long zero-sumfree sequences in a finite Abelian group. Regarding this inverse-type problem, we formulate a general conjecture and prove, among other results, that this conjecture holds true for finite cyclic groups, finite Abelian p-groups and for finite Abelian groups with rank two. Also, the results obtained here enable us to improve, via ...

The notion of a &Gamma-semigroup was introduced by Sen [8] in 1981. We can see that any semigroup can be considered as a &Gamma-semigroup. The aim of this article is to study the concept of (0-)minimal and max- imal ordered bi-ideals in ordered &Gamma-semigroups, and give some charac- terizations of (0-)minimal and maximal ordered bi-ideals in ordered &Gamma- semigroups analogous to the charact...

We study the rank of a general tensor u in a tensor product space H1 ⊗ · · · ⊗ Hk. The rank of u is the minimal number p of pure states v1, · · · , vp such that u is a linear combination of the vj ’s. This rank is an algebraic measure of the degree of entanglement of u. Motivated by quantum computation, we completely describe the rank of an arbitrary tensor in (C) and give normal forms for tens...