Prime and primary submodules of certain modules

In this paper we characterize all prime and primary submodules of the free R-module R ⁿ for a principal ideal domain R and find the minimal primary decomposition of any submodule of R ⁿ . In the case n = 2, we also determine the height of prime submodules.     

Properties of Beurling-type submodules via Agler decompositions

In this paper, we study operator-theoretic properties of the compressed shift operators $S_{z_1}$ and $S_{z_2}$ on complements of submodules of the Hardy space over the bidisk $H^2({\mathbb{D}}^2)$. Specifically, we study Beurling-type submodules – namely submodules of the form $\theta H^2({\mathbb{D}}^2)$ for θ inner – using properties of Agler decompositions of θ to deduce properties of $S_{z_1}$ and $S_{z_2}$ on model spaces $H^2({\mathbb{D}}^2)?\theta H^2({\mathbb{D}}^2)$. Results include characterizations (in terms of θ) of when a commutator $[S_{z_j}^{?},S_{z_j}]$ has rank n and when subspaces associated to Agler decompositions are reducing for $S_{z_1}$ and $S_{z_2}$. We include several open questions.     

Maps preserving a certain product of operators

Let $$\mathcal {A}$$ be a standard operator algebra on a Banach space $$\mathcal {X}$$ with $$ \dim \mathcal {X}\ge 3$$. In this paper, we determine the form of the bijective maps $$\phi :\mathcal {A}\longrightarrow \mathcal {A}$$ satisfying $$\begin{aligned} \phi \left( \frac{1}{2}(AB^2+B^2A)\right) = \frac{1}{2}[\phi (A)\phi (B)^{2}+\phi (B)^{2}\phi (A)], \end{aligned}$$for every $$A,B \in \mathcal {A}$$.     

Honest submodules

Lattices of submodules of modules and the operators we can define on these lattices are useful tools in the study of rings and modules and their properties. Here we shall consider some submodule operators defined by sets of left ideals. First we focus our attention on the relationship between properties of a set of ideals and properties of a submodule operator it defines. Our second goal will be to apply these results to the study of the structure of certain classes of rings and modules. In particular some applications to the study and the structure theory of torsion modules are provided.     

Representations of certain semidirect product groups

We study a class of semidirect product groups G = N · U where N is a generalized Heisenberg group and U is a generalized indefinite unitary group. This class contains the Poincaré group and the parabolic subgroups of the simple Lie groups of real rank 1. The unitary representations of G and (in the unimodular cases) the Plancherel formula for G are written out. The problem of computing Mackey obstructions is completely avoided by realizing the Fock representations of N on certain U-invariant holomorphic cohomology spaces.     

Hadamard product of certain starlike functions

A recent result of Shigeyoshi Owa (Math. Japon. 27 (4), (1982), 409–416) concerning the Hadamard product of starlike functions with negative coefficients is improved.     

Quasi-Hadamard product of certain univalent functions

The author improves some recent results due to Shigeyoshi Owa (Tamkang J. Math., 14 1983, 15–21) concerning the quasi-Hadamard product of certain starlike and convex univalent functions.     

Correspondences of closed submodules

If is an -faithful -module, then there is an order-preserving correspondence between the closed -submodules of and the closed -submodules of , where .

     

Hadamard product submultiplicativity of certain induced norms

In this paper we ask which norms on M_d induced by an absolute vector norm are sub-multiplicative with respect to the Hadamard product. We provide a simple necessary condition for submultiplicativity. We demonstrate that each norm on M_d induced by an l_p norm Hadamard submultiplicative and that the norms induced by certain polyhedral norms are Hadamard submultiplicative. We also consider some related inequalities.     

Hadamard product submultiplicativity of certain induced norms

In this paper we ask which norms on M_d induced by an absolute vector norm are sub-multiplicative with respect to the Hadamard product. We provide a simple necessary condition for submultiplicativity. We demonstrate that each norm on M_d induced by an l_p norm Hadamard submultiplicative and that the norms induced by certain polyhedral norms are Hadamard submultiplicative. We also consider some related inequalities.     

n-almost prime submodules

Let R be a commutative ring with identity. A proper submodule N of an R-module M will be called prime [resp. n-almost prime], if for r ∈ R and a ∈ M with ra ∈ N [resp. ra ∈ N \ (N: M) ^n?1 N], either a ∈ N or r ∈ (N: M). In this note we will study the relations between prime, primary and n-almost prime submodules. Among other results it is proved that:

1. If N is an n-almost prime submodule of an R-module M, then N is prime or N = (N: M)N, in case M is finitely generated semisimple, or M is torsion-free with dim R = 1.
2. Every n-almost prime submodule of a torsion-free Noetherian module is primary.
3. Every n-almost prime submodule of a finitely generated torsion-free module over a Dedekind domain is prime.
4. There exists a finitely generated faithful R-module M such that every proper submodule of M is n-almost prime, if and only if R is Von Neumann regular or R is a local ring with the maximal ideal m such that m ² = 0.
5. If I is an n-almost prime ideal of R and F is a flat R-module with IF ≠ F, then IF is an n-almost prime submodule of F.

     

Summand absorbing submodules of a module over a semiring

An R-module V over a semiring R lacks zero sums (LZS) if $x+y=0$ implies $x=y=0$. More generally, we call a submodule W of V “summand absorbing” (SA) in V if $?x,y\in V:x+y\in W?x\in W,y\in W$. These arise in tropical algebra and modules over idempotent semirings, as well as modules over semirings of sums of squares. We explore the lattice of finite sums of SA-submodules, obtaining analogs of the Jordan–Hölder theorem, the noetherian theory, and the lattice-theoretic Krull dimension. We pay special attention to finitely generated SA-submodules, and describe their explicit generation.