Semiprimary rings of generalized triangular type

Abstract

In the study of rings, all of whose residue rings have finite global dimension, S. Eilenberg, H. Nagao, and T. Nakayama have shown in [5] that all residue rings of a semiprimary hereditary ring have finite global dimension. One way to construct semiprimary hereditary rings is to take the ring of triangular matrices over a semisimple ring. This paper is devoted to the study of a class of semiprimary rings that contains all the rings of triangular matrices over semisimple rings. A semiprimary ring R belongs to this class (and we say that R is a T-ring) if each (left) component contains a unique minimal ideal, and each minimal (left) ideal is projective. We start by proving in Section 2 that all residue rings of a T-ring have finite global dimension. In Sections 3 and 4 we investigate the structure of T-rings. In Section 5 we discuss the (left) maximal quotient ring (in the sense of Utumi) of a T-ring, proving that it is a semisimple ring. Further properties of a T-ring R, such as the existence of a simple injective (left) R-module and a characterization of a ring of triangular matrices over a simple ring, are discussed in Section 6. In Section 7 we give some counterexamples concerning the following assertions: