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Rings whose Non-Units are a Unit Multiple of an Element from $\sqrt{Δ(R)}$

Omid Hasanzadeh, Ahmad Moussavi, Peter Danchev

Abstract

This paper introduces and studies a new class of rings called {\it $U\sqrtΔ$-rings}. A ring $R$ is $U\sqrtΔ$ if every non-unit element can be written as the product of a unit and an element from $\sqrt{Δ(R)}$, where $\sqrt{Δ(R)}$ consists of elements some power of which lies in the special subring $Δ(R)$. We establish certain basic properties of these rings and, concretely, prove that they are simultaneously indecomposable and Dedekind-finite. We also show that the polynomial ring $R[x]$ and the Laurent polynomial ring $R[x, x^{-1}]$ are never $U\sqrtΔ$-rings, while the power series ring $R[[x]]$ inherits this property from $R$. Likewise, for left (right) Artinian rings, the conditions of being a $U\sqrtΔ$-ring and a $UN$-ring are equivalent, as well as these two conditions are preserved for the full matrix ring $M_n(R)$ of size $n\geq 1$ over $R$. In addition, for a commutative ring $R$, $M_n(R)$ is a $U\sqrtΔ$-ring exactly when $R$ is local. Furthermore, we characterize when a group ring $RG$ is a $U\sqrtΔ$-ring showing that, for a locally solvable group $G$, this occurs precisely when $R$ is a $U\sqrtΔ$-ring and $G$ is a locally finite $p$-group for some prime $p \in J(R)$.

Rings whose Non-Units are a Unit Multiple of an Element from $\sqrt{Δ(R)}$

Abstract

This paper introduces and studies a new class of rings called {\it -rings}. A ring is if every non-unit element can be written as the product of a unit and an element from , where consists of elements some power of which lies in the special subring . We establish certain basic properties of these rings and, concretely, prove that they are simultaneously indecomposable and Dedekind-finite. We also show that the polynomial ring and the Laurent polynomial ring are never -rings, while the power series ring inherits this property from . Likewise, for left (right) Artinian rings, the conditions of being a -ring and a -ring are equivalent, as well as these two conditions are preserved for the full matrix ring of size over . In addition, for a commutative ring , is a -ring exactly when is local. Furthermore, we characterize when a group ring is a -ring showing that, for a locally solvable group , this occurs precisely when is a -ring and is a locally finite -group for some prime .
Paper Structure (4 sections, 31 theorems, 52 equations)

This paper contains 4 sections, 31 theorems, 52 equations.

Table of Contents

  1. Introduction and Motivations
  2. Examples and Basic Properties
  3. Matrix Rings over $U\sqrt{\Delta}$ and $UN$-rings
  4. Group Rings over $U\sqrt{\Delta}$-rings

Key Result

Lemma 2.1

For each ring $R$, we have: (1) If $a \in \sqrt{\Delta(R)}$ and $b \in U(R)$ with $ab = ba$, then $ab \in \sqrt{\Delta(R)}$. (2) If $a \in \sqrt{\Delta(R)}$, then $1 - a \in U(R)$. (3) If $a \in \sqrt{\Delta(R)} \cap C(R)$, then $a \in \Delta(R)$. (4) For every ideal $I \subseteq J(R)$, we have $\sq

Theorems & Definitions (64)

  • Lemma 2.1
  • proof
  • Remark 2.2
  • Remark 2.3
  • Proposition 2.4
  • proof
  • Theorem 2.5
  • proof
  • Example 2.6
  • Proposition 2.7
  • ...and 54 more