On Rings whose Left Modules are Direct Sums of Finitely Generated Modules

The relationship between rings of finite module type and rings whose left modules have decompositions that complement direct summands is examined by proving that the latter are precisely the rings of the title. A ring A with identity is said to be of finite module (or representation) type in case it is left artinian and has only finitely many (isomorphism types of) finitely generated indecomposable left modules. Such a ring is also right artinian and has only finitely many (the same number) finitely generated indecomposable right modules (Eisenbud and Griffith 7). Auslander 2, 3 and Ringel and Tachikawa 17 have proved that every module over a ring of finite module type is a direct sum of finitely generated modules, and Tachikawa 17 has shown that they all have decompositions M = A Ma that complement direct summands in the sense 1 that for each direct summand A of M there is a subset B E A with M = K iBMp). Chase 6 proved that the rings of the title are left artinian. (See also 8 and 11. ) More recently, Auslander 5 has proved that if A is finitely generated over its center and each left A-module is a direct sum of finitely generated modules, then A is of finite module type, and Fuller and Reiten 9 have noted that if every left and every right A-module has a decomposition that complements direct summands, then A is of finite module type. However, it is still not known whether either of these conditions on its left modules alone forces an arbitrary ring A with identity to be of finite module type. Our purpose here is to prove that nevertheless they are equivalent, and to show how their satisfaction depends on the structure of the finitely generated indecomposable left A-modules. We say that a ring A has enough idempotents in case there exists orthogonal idempotents (ea) aeA in A (called a complete set of idempotents for R) such that A = ffiAea = AeaR. By an R-module we mean an R-module with a spanning set; so "RM is a left A-module" implies M = RM = AeaM. (Note then that if 1 E A, an A-module is just a unital one. ) We denote the categories of left and right A-modules by ^'DIL and <3flL/? , and mention that they contain the regular modules RR and RR, respectively, and all ordinary submodules of their objects.