Kernels of morphisms between indecomposable injective modules.pdf (original) (raw)

KERNELS OF MORPHISMS BETWEEN INDECOMPOSABLE INJECTIVE MODULES

2010

We show that the endomorphism rings of kernels ker ϕ of non-injective morphisms ϕ between indecomposable injective modules are either local or have two maximal ideals, the module ker ϕ is determined up to isomorphism by two invariants called monogeny class and upper part, and a weak form of the Krull-Schmidt theorem holds for direct sums of these kernels. We prove with an example that our pathological decompositions actually take place. We show that a direct sum of n kernels of morphisms between injective indecomposable modules can have exactly n! pairwise non-isomorphic direct-sum decompositions into kernels of morphisms of the same type. If E R is an injective indecomposable module and S is its endomorphism ring, the duality Hom(−, E R) transforms kernels of morphisms E R → E R into cyclically presented left modules over the local ring S, sending the monogeny class into the epigeny class and the upper part into the lower part.

Direct sums of indecomposable modules

OSAKA JOURNAL OF MATHEMATICS

1. Introduction. This paper studies direct sums M=(B iζΞI M i of indecomposable modules. Specifically, we give a number of necessary and sufficient conditions for such a sum to be quasi-continuous or continuous. This question was settled in [6], in a very satisfactory way, in case the index set / is finite, or the ring is right-noetherian, but the general case dealt with here is much more complicated. Such sums M=@ i^I M i have been investigated in great detail, in a long series of papers since about 1970, by M. Harada and his collaborators, usually under the additional hypothesis that the M { have local endomorphism rings (so that the Krull-Schmidt-Azumaya Theorem applies). One of the central results is the following: Theorem 1 ([3], p. 22). For a module with a decomposition M=® ieI M h and with all endoίM,) local, the following statements are equivalent: (1) The decomposition is locally semi-T-nilpotent; (2) it complements direct summands (3) any local direct summand of M is a direct summand. (The relevent terms are defined later on in this section.

Some decomposition properties of injective and pure-injective modules

Osaka Journal of Mathematics, 1994

It is well known that over a left Notherian ring any direct sum of injecive modules is again injective, and every injective module is a direct sum of indecomposable modules. Faith introduced Σ-injective modules as modules M such that all direct sums of copies of M are injective. Cailleau showed that a Σ-injective module is a direct sum of indecomposable modules. The concept of Σ-injective modules had several intersting developments and applications (see e.g. Faith ). Also, some generalizations of Σ-injective modules, such as Σ-quasi-injective modules (Cailleau-Renault [3]), or Σ-M-injecitve modules (Albu-Nastasescu [1]), were studied. Of special interest are Σ-pure-injective modules which were introduced and investigated extensively by W. Zimmermann and B. Zimmermann-Huisgen . These modules include, besides Σ-injective modules, also Π-projective modules (i.e., modules M such that all direct products of copies of M are projective).

Direct Sums of Infinitely Many Kernels

Journal of the Australian Mathematical Society, 2010

Let K be the class of all right R-modules that are kernels of nonzero homomorphisms ϕ : E 1 → E 2 for some pair of indecomposable injective right R-modules E 1 , E 2 . In a previous paper, we completely characterized when two direct sums A 1 ⊕ · · · ⊕ A n and B 1 ⊕ · · · ⊕ B m of finitely many modules A i and B j in K are isomorphic. Here we consider the case in which there are arbitrarily, possibly infinitely, many A i and B j in K. In both the finite and the infinite case, the behaviour is very similar to that which occurs if we substitute the class K with the class U of all uniserial right R-modules (a module is uniserial when its lattice of submodules is linearly ordered).

A note on indecomposable modules

Rendiconti del Circolo Matematico di Palermo, 1988

In this note we study rings having only a finite number of non isomorphic uniform modules with non zero socle. It is proved that a commutative ring with this property is a direct sum of a finite ring and a ring of finite representation type. In the non commutative case we show that most P.I. rings having only a finite number of non isomorphic modules with non zero socle are in fact artinlan.

MULTIPLICITIES OF INDECOMPOSABLE INJECTIVES

Journal of The London Mathematical Society-second Series, 2005

Several results about the multiplicities of indecomposable injectives in the minimal injective resolution of a ring exist in the literature. Mostly these apply to universal enveloping algebras of finite dimensional solvable Lie algebras, and to Gorenstein noetherian PI local rings. We unify these results and extend them to the much wider class of rings with Auslander dualizing complexes.