Almost M-projectives and nakayama rings

Abstract

In 1960, H. Bass [3] defined the notion of semi-perfect rings and, in 1963, E. Mares [l l] generalized it to that of semi-perfect modules. Since that time various kinds of further generalizations have been considered. In particular, K. Oshiro [ 161 studied quasi-semiperfect modules in connection with several important conditions on direct summands. The present paper is closely related to his paper as well as to the papers [ 12, 151 by Mueller and others and is cncerned with the condition (D,) (which is (C,) in [16]) over modules with direct decomposition into hollow modules. This is indeed motivated by the theorem [ 16, Theorem 3.51 that every quasisemiperfect module has the above type of decomposition. In Section 2 we shall clarify relationships between M-projectives over modules with direct decomposition into hollow modules and the condition (D,), which is dual to Cl.5, Theorem 81. In order to translate (D, ) in terms of homomorphisms, we shall introduce a new concept of almost A4-projectives to find some relations between almost M-projectives and a weaker condition (D’,). After preparing several basic results in Section 3, we shall, in Section 4, restrict ourselves to semi-perfect rings and give a series of characterizations of right Nakayama rings with special properties by making use of the concept of almost M-projectives. Finally, we shall classify modules with (D,) over a local Dedekind domain in Section 5.