# Elementary Set Theory by Kam-Tim Leung

By Kam-Tim Leung

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Skowrofiski: On some classes of simply connected algebras, Proc. London Math. Soc. (3)56 (1988) 417-450. [9] I. Assemand A. Skowrofiski: Algebras with cycle-finite Math. Ann. 280 (1988) 441-463. derived categories, [10] I. Assemand A. Skowroxiski: Quadratic forms and iterated tilted Algebra 128 (1990) 55-85. algebras, J. Strongly SimplyConnectedDerivedTubularAlgebras 29 [11] K. Bongartz and P. Gabriel: Covering spaces in representation theory, Invent. Math. 65 (1981) 331-378. [12]O. Bretscher and P.

ThenA is a finitely generatedleft (right) A/~-module. Proo]. Let G be the set of group-like elements in H. Then G is a group and a~,... , a,~ E G. Thegroup algebra kG is a subalgebrain H. HenceG is finite and a~,... , an have finite orders. 1]. THEOREM 5. Let H,p,A,B,A~,~ be from Theorem 3. ThenA is a finitely generated]eft (right) moduleover the subalgebra of coinvariants AH. Proof. By Corollary 1 there exists a positive integer d such that (22) holds. f~ = X~ + X~-d. Since )~t~ = 0 wehave -~ = ~(~) = ~(x~)~ + a(x~) [~ ® X~ + m ® X - i-~] ~ + [:~7 ~ ® X~-~ + ~-~ ® X~]~ = (~ + ,;-~) ~ x - ~ + (~7~ + ~) ®x7~ = (~ + ~) ®x~+ (1~+ 1~)-~= 1 ® ~.

Emodules. Further, we know from [8, Theorem 7] or [10] that R is right puresemisimple if and only if so is E, which means by [12, Theorem 6] that every pure-projective left E-module is endonoetherian, or in other words, that the left E-module I-[ie~ Hom~(Mi,W) is endonoetherian. 3 the latter is equivalent to ILe~ Hom~(Mi,W) ~ ~M*being endonoetherian. 6 the proof is complete. (2) Take Mas in (1). 3 we knowthat every finitely presented indecompos;~ble module ARhas a generalized left almost split morphism if and only if Mis endonoetherian.