Simple module

About: Simple module is a research topic. Over the lifetime, 1924 publications have been published within this topic receiving 24070 citations.

Injective modules over Noetherian rings.

Abstract: Introduction In this discussion every module over a ring R will be understood to be a left i2-module. R will always have a unit, and every module will be unitary. The aim of this paper is to study the structure and properties of injective modules, particularly over Noetherian rings. B. Eckmann and A. Schopf have shown that if M is a module over any ring, then there exists a unique, minimal, injective module E(M) containing it. The module E(M) will be a major tool in our investigations, and we shall systematically exploit its properties. In § 1 we show that if a module M has a maximal, injective submodule C (as is the case for left-Noetherian rings), then C contains a carbon-copy of every injective submodule of M, and MjC has no injective submodules different from 0. Although C is unique up to an automorphism of My C does not in general contain every injective submodule of M, In fact, the sum of two injective submodules of a module is always injective if and only if the ring is left-heredita ry. In § 2 we show that for any ring R a module E is an indecomposable, injective module if and only if E = E(R\J)y where J is an irreducible, left ideal of R. We prove that if R is a left-Noetherian ring, then every injective Jϋ-module has a decomposition as a direct sum of indecomposable, injective submodules. Strong uniqueness assertions can be made concerning such decompositions over any ring. In § 3 we take R to be a commutative, Noetherian ring, and P to be a prime ideal of R. We prove there is a one-to-one correspondence between the prime ideals of R and the indecomposable, injective Rmodules given by P**E(RjP). We examine the structure of the module E = E{RjP)y and show that if At is the annihilator in E of P\ then E = U At and -4ί+1/At is a finite dimensional vector space over the quotient field of R/P. The ring of iϋ-endomorphi sms of E is isomorphic in a natural way to Rp, the completion of the ring of quotients of R with respect to R-P. As an ^-module E is an injective envelope of RpjP, where P is the maximal ideal of Rp. If P is a maximal ideal of Ry then E is a countably generated β-module. Every indecomposable, injective i2-module is finitely generated if and only if R has the minimum condition on ideals. In § 4 we take R to be a commutative, Noetherian, complete, local ring, P the maximal ideal of R and E = E{RjP). Then the eontravariant,

Hall algebras, hereditary algebras and quantum groups.

Abstract: Let R be an associative, hereditary algebra over a finite field k, and let R-fin be the full subcategory of R-mod whose objects are those left R-modules X which are finite as sets, IX] < oc. Assume also that R isfinitary in C. Ringel's sense, i.e. that IExt l (s , s ' ) ] < c~ for all simple S,S' in R-fin; this condition is met, for example, if R is finitely generated as k-algebra [4, pp. 435,436]. Let ~ be the set of all isomorphism classes in R-fin. If 2 E ~ , then U~ will denote an R-module in class 2. The class of all zero left R-modules is denoted 0. Let I C_ ~ be the set of all isomorphism classes of simple modules in R-fin. Thus {U~ : i E I} is a complete set of simple, finite left R-modules. We identify the Grothendieck group K0(R-fin) with the free Abelian group E1 = {~ , v~i : v~ E 7Z} having I as free basis, so that if X E R-fin then the corresponding element of K0(R-fin) is the "dimension vector" dimX = Y~,~I vii, where for each i E /, vi is the multiplicity of the simple module U, in any composition series of X. Clearly d imX lies in the subsemigroup