Projective resolutions and quasi-heredity (Q757555)

Let A be an artinian ring. \textit{E. Green} defined [Representation Theory II, Lect. Notes Math. 832, 259-279 (1980; Zbl 0491.16015)] the type of A via the existence of a partial ordering of the set of isomorphism classes of indecomposable projective A-modules, such that the ordering satisfies conditions given by certain projective resolutions. In this note, the connection between a slightly modified version of Green's definition and the concept of quasi-heredity [\textit{E. Cline, B. Parshall} and \textit{L. L. Scott}: J. Reine Angew. Math. 391, 85-99 (1988; Zbl 0657.18005)] is studied. It is shown, that an artinian ring of type at most one is quasi- hereditary. This generalizes a result of \textit{V. Dlab} and \textit{C. M. Ringel} [Ill. J. Math. 33, 280-291 (1989; Zbl 0666.16014)] that global dimension two implies quasi-heredity. Moreover, endomorphism rings of certain projective modules over an algebra A of type at most 1 are studied and it is shown, that chains of such endomorphism rings can be chosen with all rings in the chain again being of type at most 1 and having global dimension less than or equal to gldim(A). To all definitions and results, analogues for orders over discrete valuation rings are given, using the author's concept of quasi-hereditary orders [Manuscr. Math. 68, 417-433 (1990; Zbl 0706.16008)].