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.Comment: To appear in Glassgow Mthematical Journal, 12 page
