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Nil-Essential Ideals

Raplang Nongsiej, Ardeline Mary Buhphang

Abstract

The class of nil-essential ideals is a generalisation of the class of essential ideals. Every nil-essential ideal of a reduced ring is essential. Therefore the intersection of all nil-essential ideals over a reduced ring $R$ is the socle of $R$. In this note, we apply this generalisation to give a new criteria of semisimplicity in terms of nil-essentiality of ideals.

Nil-Essential Ideals

Abstract

The class of nil-essential ideals is a generalisation of the class of essential ideals. Every nil-essential ideal of a reduced ring is essential. Therefore the intersection of all nil-essential ideals over a reduced ring is the socle of . In this note, we apply this generalisation to give a new criteria of semisimplicity in terms of nil-essentiality of ideals.
Paper Structure (4 sections, 19 theorems, 3 equations)

This paper contains 4 sections, 19 theorems, 3 equations.

Table of Contents

  1. Introduction
  2. Nil-Essential Ideals
  3. Nil-Essential Monomorphisms
  4. Localisation of Nil-Essential Ideals

Key Result

Proposition 1

Let $R$ be a ring. If a left(resp. right) ideal $I$ is nil-essential, then every left(resp. right) ideal of $R$ containing $I$ is nil-essential.

Theorems & Definitions (32)

  • Proposition 1
  • Corollary 2
  • Corollary 3
  • Corollary 4
  • Corollary 5
  • Proposition 6
  • proof
  • Proposition 7
  • proof
  • Corollary 8
  • ...and 22 more