Ring Endomorphisms with Large Images

TL;DR

This paper introduces the concept of ring endomorphisms with large images, studying their injectivity and surjectivity, and establishing conditions under which such endomorphisms are automorphisms, with examples and open questions.

Contribution

It defines large image endomorphisms and proves injectivity and automorphism conditions for prime noetherian rings, expanding understanding of endomorphism properties.

Findings

Injectivity of endomorphisms when the image contains an essential left ideal.

Endomorphisms are automorphisms if they fix an essential left ideal.

Examples show assumptions cannot be weakened to prime left Goldie rings.

Abstract

The notion of ring endomorphisms having large images is introduced. Among others, injectivity and surjectivity of such endomorphisms are studied. It is proved, in particular, that an endomorphism S of a prime one-sided noetherian ring R is injective whenever the image S (R) contains an essential left ideal L of R. If additionally S(L) = L, then S is an automorphism of R. Examples showing that the assumptions imposed on R can not be weakened to R being a prime left Goldie ring are provided. Two open questions are formulated.