\({\mathfrak U}\)-fibre sums in Artinian rings. (Q1822596)

Let A-mod be the category of finitely generated left modules over an artinian ring A and suppose that U is a module in A-mod such that \(Ext^ 1_ A(U,X)=0\) for all X in (Sub U)\(\cup (Fac U)\), where Sub U (resp. Fac U) denotes the full subcategory of A-mod consisting of submodules (resp. factor modules) of finite direct sums of U. Denote by \({\mathcal I}\) the ideal in A-mod consisting of homomorphisms having a factorization through modules Y such that \(Hom_ A(U,Y)=0\) and suppose that there are only finitely many indecomposable modules \(W_ 1,...,W_ n\) in A-mod/\({\mathcal I}\) such that \(Ext^ 1_ A(W_ i,X)=0\) for all X in Fac U. It is proved in the paper that if A-mod/\({\mathcal I}\) is representation finite and \(B=End(V)/{\mathcal I}(V,V)\), \(V=W_ 1\oplus...\oplus W_ n\), then A- mod/\({\mathcal I}\) is equivalent to the subcategory \(B\)-mod\({}_ e\) of B-mod consisting of modules Z such that \(Ann_ Z(BeB)=0\), where e is the idempotent in B corresponding to the direct sum of all indecomposable summands of U. In particular, if U is indecomposable, A is a PI-ring and \(End(W_ i)/{\mathcal I}(W_ i,W_ i)\) is a division ring for \(i=1,...,n\), then B is a Schurian right peak PI-ring and \(B\)-mod\({}_ e\) is the category of socle projective left B-modules studied by the reviewer [in J. Algebra 92, 532-571 (1985; Zbl 0558.16016)].