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Modules over linear spaces admitting a multiplicative basis

Antonio J. Calderón, Francisco J. Navarro Izquierdo, José M. Sánchez

Abstract

We study the structure of certain modules $V$ over linear spaces $W$ with restrictions neither on the dimensions nor on the base field $\mathbb F$. A basis $\mathfrak B = \{v_i\}_{i\in I}$ of $V$ is called multiplicative respect to the basis $\mathfrak B' = \{w_j\}_{j \in J}$ of $W$ if for any $i \in I, j \in J$ we have either $v_iw_j = 0$ or $0 \neq v_iw_j \in \mathbb Fv_k$ for some $k \in I$. We show that if $V$ admits a multiplicative basis then it decomposes as the direct sum $V=\bigoplus_k V_k$ of well-described submodules admitting each one a multiplicative basis. Also the minimality of $V$ is characterized in terms of the multiplicative basis and it is shown that the above direct sum is by means of the family of its minimal submodules, admitting each one a multiplicative basis.

Modules over linear spaces admitting a multiplicative basis

Abstract

We study the structure of certain modules over linear spaces with restrictions neither on the dimensions nor on the base field . A basis of is called multiplicative respect to the basis of if for any we have either or for some . We show that if admits a multiplicative basis then it decomposes as the direct sum of well-described submodules admitting each one a multiplicative basis. Also the minimality of is characterized in terms of the multiplicative basis and it is shown that the above direct sum is by means of the family of its minimal submodules, admitting each one a multiplicative basis.
Paper Structure (3 sections, 9 theorems, 4 equations)

This paper contains 3 sections, 9 theorems, 4 equations.

Table of Contents

  1. Introduction and previous definitions
  2. Connections in the set of indexes. Decompositions
  3. The minimal components

Key Result

lemma 1

Let $a,b\in I$ be. Given $j\in J\;\dot\cup\;\overline{J}$ we have that $a\in b\star j$ if and only if $b\in a\star \overline{j}$.

Theorems & Definitions (24)

  • definition 1
  • definition 2
  • remark 1
  • lemma 1
  • proof
  • lemma 2
  • proof
  • definition 3
  • lemma 3
  • proof
  • ...and 14 more