Affirmative answer to the Question of Leroy and Matczuk on injectivity of endomorphisms of semiprime left Noetherian rings with large images

TL;DR

The paper proves that all endomorphisms with large images of semiprime left Noetherian rings are injective, affirmatively answering Leroy and Matczuk's question and extending the result to broader classes of rings.

Contribution

It establishes that endomorphisms with large images of semiprime left Noetherian rings are necessarily injective, resolving an open question and generalizing previous results.

Findings

Endomorphisms with large images are either injective or have kernels containing regular elements.

All such endomorphisms of semiprime rings with Krull dimension are injective.

The main result confirms the conjecture for semiprime left Noetherian rings.

Abstract

The class of semiprime left Goldie rings is a huge class of rings that contains many large subclasses of rings -- semiprime left Noetherian rings, semiprime rings with Krull dimension, rings of differential operators on affine algebraic varieties and universal enveloping algebras of finite dimensional Lie algebras to name a few. In the paper, `Ring endomorphisms with large images,' {\em Glasg. Math. J.} {\bf 55} (2013), no. 2, 381--390, A. Leroy and J. Matczuk posed the following question: {\em If a ring endomorphism of a semiprime left Noetherian ring has a large image, must it be injective?} The aim of the paper is to give an affirmative answer to the Question of Leroy and Matczuk and to prove the following more general results. {\bf Theorem. (Dichotomy)} {\em Each endomorphism of a semiprime left Goldie ring with large image is either a monomorphism or otherwise its kernel contains a regular element of the ring ($\Leftrightarrow$ its kernel is an essential left ideal of the ring). In general, both cases are non-empty.} {\bf Theorem. } {\em Every endomorphism with large image of a semiprime ring with Krull dimension is a monomorphism.} {\bf Theorem. (Positive answer to the Question of Leroy and Matczuk)} {\em Every endomorphism with large image of a semiprime left Noetherian ring is a monomorphism.}