Mh. Math. 109, 327--331 (1990) ~ ~ t i i r ' " Malhomalik 9 by Springer-Verlag 1990 An Application of a Theorem of Ziegler By Maher Zayed, Banha (Received 3 November 1989) Abstract. The aim of this note is to prove that an artin algebra (resp. a finite dimensional algebra over an algebraically closed field) all of whose algebraically compact (resp. E-algebraically compact) indecomposable modules are finitely generated must be of finite-representation type. The result is derived with the aid of a theorem of Ziegler. I. Preliminaries The global conventions are: A denotes an artin algebra, "modules" mean left A-modules. Let us recall that an exact sequence of A-modules 0 "-'~ M 1 - - ~ M ~ M 2 - ~ O is called pure-exact if the sequence 0 ~ Hom A(F, M1) ~ Homn (F, M) ~ HomA (F, M2) ~ 0 is exact for each finitely presented A-module F. An A-module X is called algebraically compact (= pure-injective) if every pure-exact sequence of the form 0 ~ X ~ Y--. Z ~ 0 is split exact. An A-module X is called E-algebraically compact or Z-pure-injective if each direct sum X (I) is algebraically compact. An A-module P is called pure-projec- tive if every pure-exact sequence of the form 0 ~ N~ M~ P ~ 0 splits. Since A is artinian, the pure-projective modules are exactly the direct sums of finitely presented ones, as follows from a theorem of Crawley--Johnsson--Warfield [1, Th. 26.5]. A long exact sequence ...-~ M,_ 1-~ M~ I"--~ M~ + 1-, ... is called pure-exact iff, (Mn_ ~) = Ker(s ~) is a pure submodule of 214, for each n.