scholarly journals Direct summands of direct sums of modules whose endomorphism rings have two maximal right ideals

2011 ◽  
Vol 215 (9) ◽  
pp. 2209-2222 ◽  
Author(s):  
Afshin Amini ◽  
Babak Amini ◽  
Alberto Facchini
Keyword(s):  
Endomorphism Rings ◽  
Direct Sums ◽  
Direct Sums Of Modules

scholarly journals Direct Sums of Representations as Modules over Their Endomorphism Rings

2002 ◽  
Vol 250 (1) ◽  
pp. 67-89 ◽  
Author(s):  
Birge Huisgen-Zimmermann ◽  
Manuel Saorín
Keyword(s):  
Endomorphism Rings ◽  
Direct Sums

A Remark on the Krull-Schmidt-Azumaya Theorem

1970 ◽  
Vol 13 (4) ◽  
pp. 501-505 ◽  
Author(s):  
B. L. Osofsky
Keyword(s):  
Dedekind Domain ◽  
The Other ◽  
Group Algebras ◽  
Endomorphism Rings ◽  
Algebraic Integers ◽  
Direct Sums ◽  
Indecomposable Modules ◽  
Direct Sum ◽  
Other Hand

It is well known that if a module M is expressible as a direct sum of modules with local endomorphism rings, then such a decomposition is essentially unique. That is, if M = ⊕i∊IMi = ⊕j∊JNj then there is a bijection f: I → J such that Mi is isomorphic to Nf(i) for all i∊I (see [1]). On the other hand, a nonprincipal ideal in a Dedekind domain provides an example where such a theorem fails in the absence of the local hypothesis. Group algebras of certain groups over rings R of algebraic integers is another such example, where even the rank as R-modules of indecomposable summands of a module is not uniquely determined (see [2]). Both of these examples yield modules which are expressible as direct sums of two indecomposable modules in distinct ways. In this note we construct a family of rings which show that the number of summands in a representation of a module M as a direct sum of indecomposable modules is also not unique unless one has additional hypotheses.

Abelian Groups Quasi-Injective Over their Endomorphism Rings

1972 ◽  
Vol 24 (4) ◽  
pp. 617-621 ◽  
Author(s):  
George D. Poole ◽  
James D. Reid
Keyword(s):  
Endomorphism Ring ◽  
Abelian Groups ◽  
Invariant Subgroup ◽  
Injective Module ◽  
Endomorphism Rings ◽  
Additive Structure ◽  
Direct Sums ◽  
Divisible Groups

L. Fuchs has posed the problem of identifying those abelian groups that can serve as the additive structure of an injective module over some ring [1, p. 179], and in particular of identifying those abelian groups which are injective as modules over their endomorphism rings [1, p. 112]. Richman and Walker have recently answered the latter question, generalized in a non-trivial way [7], and have shown that the groups in question are of a rather restricted structure.In this paper we consider abelian groups which are quasi-injective over their endomorphism rings. We show that divisible groups are quasi-injective as are direct sums of cyclic p-groups. Quasi-injectivity of certain direct sums (products) is characterized in terms of the summands (factors). In general it seems that the answer to the question of whether or not a group G is quasinjective over its endomorphism ring E depends on how big HomE(H, G) is, with H a fully invariant subgroup of G.

scholarly journals KERNELS OF MORPHISMS BETWEEN INDECOMPOSABLE INJECTIVE MODULES

2010 ◽  
Vol 52 (A) ◽  
pp. 69-82 ◽  
Author(s):  
ALBERTO FACCHINI ◽  
ŞULE ECEVIT ◽  
M. TAMER KOŞAN
Keyword(s):  
Local Ring ◽  
Endomorphism Ring ◽  
Weak Form ◽  
Indecomposable Module ◽  
Endomorphism Rings ◽  
Direct Sums ◽  
Indecomposable Modules ◽  
Direct Sum ◽  
Injective Modules ◽  
Maximal Ideals

AbstractWe 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 ofnkernels of morphisms between injective indecomposable modules can have exactlyn! pairwise non-isomorphic direct-sum decompositions into kernels of morphisms of the same type. IfERis an injective indecomposable module andSis its endomorphism ring, the duality Hom(−,ER) transforms kernels of morphismsER→ERinto cyclically presented left modules over the local ringS, sending the monogeny class into the epigeny class and the upper part into the lower part.

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