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Graded pseudo-H-rings

Antonio J. Calderón, Antonio Díaz, Marina Haralampidou, José M. Sánchez

Abstract

Consider a pseudo-$H$-space $E$ endowed with a separately continuous biadditive associative multiplication which induces a grading on $E$ with respect to an abelian group $G$. We call such a space a graded pseudo-$H$-ring and we show that it has the form $E = cl(U + \sum_j I_j)$ with $U$ a closed subspace of $E_1$ (the summand associated to the unit element in $G$), and any $I_j$ runs over a well described closed graded ideal of $E$, satisfying $I_jI_k = 0$ if $j \neq k$. We also give a context in which graded simplicity of $E$ is characterized. Moreover, the second Wedderburn-type theorem is given for certain graded pseudo-$H$-rings.

Graded pseudo-H-rings

Abstract

Consider a pseudo--space endowed with a separately continuous biadditive associative multiplication which induces a grading on with respect to an abelian group . We call such a space a graded pseudo--ring and we show that it has the form with a closed subspace of (the summand associated to the unit element in ), and any runs over a well described closed graded ideal of , satisfying if . We also give a context in which graded simplicity of is characterized. Moreover, the second Wedderburn-type theorem is given for certain graded pseudo--rings.
Paper Structure (3 sections, 9 theorems, 56 equations)

This paper contains 3 sections, 9 theorems, 56 equations.

Table of Contents

  1. Introduction and Preliminaries
  2. Decompositions
  3. The graded simple components

Key Result

Lemma 1.4

Let $E$ be a topological ring. If, in addition $E$ is Hausdorff and complete, then the following hold:

Theorems & Definitions (27)

  • Example 1.1
  • Definition 1.2
  • Example 1.3
  • Lemma 1.4
  • proof
  • Definition 2.1
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • ...and 17 more