A generalization of a theorem of M. Auslander. (English) Zbl 0669.16018

Bongartz, Klaus

Bull. Lond. Math. Soc. 21, No. 3, 255-256 (1989).

The note contains an elementary proof of the following generalization of a result of M. Auslander [Contemp. Math. 13, 27-39 (1982; Zbl 0529.16020)]. Let R be a commutative ring and A an abelian R-linear category such that each morphism set in A has finite length as an R- module. Let C be a full subcategory of A closed under direct sums and kernels. Then two objects M and N of C are isomorphic if and only if the lengths of Hom(M,X) and Hom(N,X) coincide for all X in C.

Reviewer: A.Skowroński (Toruń)

MSC:

16P10	Finite rings and finite-dimensional associative algebras
16Gxx	Representation theory of associative rings and algebras
16B50	Category-theoretic methods and results in associative algebras (except as in 16D90)

Keywords:

abelian R-linear category; morphism set; finite length; full subcategory