Coherence and Generalized Morphic Property of Triangular Matrix Rings

Let R be a ring. R is left coherent if each of its finitely generated left ideals is finitely presented. R is called left generalized morphic if for every element a in R, l(a) = Rb for some b ∈ R, where l(a) denotes the left annihilator of a in R. The main aim of this article is to investigate the coherence and the generalized morphic property of the upper triangular matrix ring T _n (R) (n ≥ 1). It is shown that R is left coherent if and only if T _n (R) is left coherent for each n ≥ 1 if and only if T _n (R) is left coherent for some n ≥ 1. And an equivalent condition is obtained for T _n (R) to be left generalized morphic. Moreover, it is proved that R is left coherent and left Bézout if and only if T _n (R) is left generalized morphic for each n ≥ 1.