Coherence and Generalized Morphic Property of Triangular Matrix Rings

Abstract

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.

Key Words:

• Bézout ring
• Coherent ring
• Generalized morphic ring
• Upper triangular matrix ring

2010 Mathematics Subject Classification:

• 16S99
• 16P70
• 16U99

ACKNOWLEDGMENTS

This research is supported in part by the National Natural Science Foundation of China (11371089), the Specialized Research Fund for the Doctoral Program of Higher Education (No. 20120092110020), the National Natural Science Foundation of China (No. 11201064), the Natural Science Foundation of Jiangsu Province (BK20130599), and the National Natural Science Foundation of China (No. 11226071).

Notes

Communicated by Q. Wu.