Kunio Yamagata

Extensions over hereditary Artinian rings with self-dualities, I

In this paper we study the finitely generated indecomposable modules over an arbitrary extension over an Artinian ring with self-(Morita) duality. Let A be an Artinian ring with self-duality and T an extension over A with kernel Q (see [ 11, Chap. XIV, Sect. 21) such that QA and A Q are isomorphic to injective hulls of top(A,) and topLA), respectively. Such an A-module Q will be called quasi-Frobenius. Then it will be proved that

On artinian rings of finite representation type

In [5], Roiter has solved the Brauer-Thrall conjecture for finite-dimensiona algebras over fields, which states that, if the lengths of the finitely generated indecomposable modules are bounded, then there is only a finite number of finitely generated indecomposablc modules. Rcccntly Auslander [l] has proved it for ilrtinian rings. In this paper, we show how we construct all indecomposable modules from simple modules over an Artinian ring of finite representation type. That is, if A is an Artinian ring of finite representation type, then every indecomposable A-module appears as a direct summand of (a) the radical of a projective indecomposable A-module or (b) the middle term of an almost split sequence [2], which is successively obtained from a simple ,4-module. Simultaneously, we give a module-theoretical, self-contained, and simple proof for the conjecture though Auslander’s proof is categorical. Throughout this paper ;I will be a right Artinian ring with identity and all modules will be finitely generated right A-modules. Let M be an indecomposable module and N a module. Following Auslander [ 11, a homomorphism f : N ---+ M is said to be aZmst splittable if (a) it is not a splittable epimorphism and (b) for any homomorphism g: S + M which is not a splittable epimorphism, there is a homomorphism h: S -+ N such that g == fh. In the following, an almost splittable homomorphism f : N --+ JZ will be called almost split extemion ozw AI provided that (a) if JZ is projective, then N is the unique maximal submodule of A!! andfis the inclusion, or (b) if M is not project&c, thenfis an epimorphism and Ker f is indecomposable, in which case 0 -+ KerfN f, AI --z 0 is called ahost split sequence in the sense of [2]. It is known [I, 21 that an almost split extension is uniquely determined up to isomorphism and that if the ring A is an Artin algebra or is of finite representation type, then there is an almost split extension over any indecomposable A-module. But it is an open question whethet almost split extensions always exist for arbitrary right Artinian rings. In the following, [M] denotes the isomorphism class of a given module M. For an indecomposable module M, we define a set E,(M) (n 2 0) of finitely many isomorphism classes of indecomposable modules as follows:

Representations of non-splittable extension algebras

Let A be an algebra over a field K and D a self-duality D: mod A 2 mod AoP, where AoP is the opposite algebra of A and mod A the category of finitely generated left A-modules. By an extension for short we understand an extension algebra T over A with kernel DA in the sense of Cartan-Eilenberg [3]: 0 -+ DA + T -+p A --+ 0, where p is an algebra epimorphism. In the case where A is hereditary, by means of the Heller function QAXDA, the Auslander-Reiten quiver rAwDA of the trivial extension A K DA is completely determined by rA [lo]. Moreover, for any extension T, Tr is isomorphic to rAwDA. It seems that this fact suggests the existence of a closer connection between some categories of modules over T and over A 1x DA, though mod T is not in general equivalent to mod A K DA. In this paper we are concerned with categorical relations between non-splittable extensions and the trivial extensions, and we shall establish some relation between them for some class of algebras to which hereditary algebras belong. We prove the following, where mod, A denotes the category of finitely generated left A-modules without projective summands.

Modules with serial Noetherian endomorphism rings

In this paper we shall characterize the generators whose endomorphism rings are serial Noetherian. A module is serial when its submodules are linearly ordered with respect to inclusion. A ring L! is said to be right serial if A,, is a direct sum of serial submodules, and ,4 is serial if it is both left and right serial. By Warlield’s structure theorem [7], a serial Noetherian ring /i is the product of a serial Artinian ring and a finite number of serial prime Noetherian rings. We denote by A, the Artinian component and A,, the Noetherian component of a serial Noetherian ring A; n = A, x A,. Given a module M, by add(M) we understand the category of modules isomorphic to summands of finite direct sums of M. Our aim of this paper is to give a simple characterization for a module over a serial Artinian ring to have the serial endomorphism ring, and to prove the following theorem.

A note on Hughes-Waschbüsch reflections of algebras

Abstract Let A be a finite-dimensional algebra over an algebraically closed field. A section S of the Auslander-Reiten quiver Γ ( A ) of A is called a left strong section when it does not have injective predecessers (except in S ) and every source of S is injective. With a sink i of the Gabriel quiver of A , an algebra S + i A is associated, which naturally induces a partial map from Γ ( A ) to Γ ( S + i A ).In this paper we investigate the map for simply connected algebras, and prove that the number of left strong sections of S + i A is not greater than the number of left strong sections of A .