ScholarLens Back to search
1978•Canadian Journal of MathematicsOpen access

A Condition for Artinian Rings to be Noetherian

Ichiro Murase

Open full text 3 citations

Abstract

Throughout this paper the word “Artinian (Noetherian) ring” means an associative ring with minimum (maximum) condition on left ideals. According to C. Hopkins, an Artinian ring is Noetherian if it contains a left or right identity [3, p. 728]. However we shall consider Artinian rings without the assumption of existence of such an identity, and the theorem of Hopkins will be reproved.

Open-access reader

About this research paper

What this paper is about

Throughout this paper the word “Artinian (Noetherian) ring” means an associative ring with minimum (maximum) condition on left ideals. According to C. Hopkins, an Artinian ring is Noetherian if it contains a left or right identity [3, p. 728]. However we shall consider Artinian rings without the assumption of existence of such an identity, and the theorem of Hopkins will be reproved.

Why it matters

OpenAlex reports 3 citations for this work. Citation counts describe recorded attention and do not establish research quality.

Key contribution

A contribution statement is not available in the OpenAlex record.

Method / approach

Method details are not available in the OpenAlex metadata.

Main findings

Findings are not separately available in the OpenAlex metadata.

Limitations

Limitations are not available in the OpenAlex metadata.

Applications

Application details are not available in the OpenAlex metadata.

Available abstract

Throughout this paper the word “Artinian (Noetherian) ring” means an associative ring with minimum (maximum) condition on left ideals. According to C. Hopkins, an Artinian ring is Noetherian if it contains a left or right identity [3, p. 728]. However we shall consider Artinian rings without the assumption of existence of such an identity, and the theorem of Hopkins will be reproved.

Key concepts: Mathematics, Artinian ring, Noetherian, Semisimple module, Noncommutative ring, Radical of a ring, Pure mathematics, Principal ideal ring

Related papers

Related paper 1Related paper 2Related paper 3
Back to paper searchBrowse research topicsOriginal source
A Condition for Artinian Rings to be Noetherian — Research Paper | ScholarLens