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Schur rings over Free Abelian Group of Rank Two

Gang Chen, Jiawei He, Zhiman Wu

Abstract

Schur rings are a type of subrings of group rings afforded by a partition of the underlined group. In this paper, Schur rings over free abelian group of rank two are classified under the assumption that one of the direct factor is a union of some basic sets. There are eight different types, and all but one type of which are traditional.

Schur rings over Free Abelian Group of Rank Two

Abstract

Schur rings are a type of subrings of group rings afforded by a partition of the underlined group. In this paper, Schur rings over free abelian group of rank two are classified under the assumption that one of the direct factor is a union of some basic sets. There are eight different types, and all but one type of which are traditional.
Paper Structure (6 sections, 12 theorems, 95 equations)

This paper contains 6 sections, 12 theorems, 95 equations.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Schur Rings
  4. Traditional Schur rings
  5. Proof of the main Results
  6. Acknowledgment

Key Result

Theorem 1.1

Let $\mathcal{A}$ be a Schur ring over $\mathcal{Z}\times\mathcal{Z}$. If there exists $k\in \mathbb{Z}^{\#}$ such that $\langle a^k\rangle$ is an $\mathcal{A}$-subgroup. Then one the following holds: Furthermore, all but the last type is traditional.

Theorems & Definitions (13)

  • Theorem 1.1
  • Definition 2.1
  • Lemma 2.1
  • Theorem 2.1
  • Proposition 2.1
  • Corollary 2.1
  • Proposition 2.2
  • Lemma 2.2
  • Theorem 2.2
  • Lemma 3.1
  • ...and 3 more