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

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 ($\iff$ 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.}