A Splitting Criterion for a Class of Mixed Modules

Mutzbauer, Otto; Toubassi, Elias

doi:10.1216/rmjm/1181072355

Cited by 9 publications

(8 citation statements)

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“…[6,7,8,9]. As in [6] we denote by h 1 the subclass of h consisting of the modules of torsion-free rank one.…”

Section: Preliminariesmentioning

confidence: 99%

“…As in [6] we denote by h 1 the subclass of h consisting of the modules of torsion-free rank one. We now recall from [9] the definition of a module by generators and relations. Let G be a module in the class h of torsion-free rank d with torsion submodule tG v iPN v vPIi Rx v i of isomorphism type l s i j i P N, where s i jI i j and ann x v i p i R for i P NY v P I i .…”

Section: Preliminariesmentioning

confidence: 99%

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Characterizing a class of Warfield modules by Ulm submodules and Ulm factors

2001

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Warfield modules and simply presented modules are considered whose torsion submodule is a direct sum of cyclics. A correspondence between Warfield modules and completely decomposable groups is established by showing that such a mixed module G is Warfield if and only if Gap w G is simply presented and p w G is completely decomposable. We give an example to show that this result cannot be extended to ordinals s b w. We also connect with the work of Nunke to characterize modules G whose zeroth Ulm factors are torsion in terms of Gap s G simply presented and p s G completely decomposable.Introduction. In this paper we investigate Warfield modules and simply presented modules whose torsion submodules are a direct sum of cyclics. We show a correspondence between Warfield modules and completely decomposable modules by showing that such mixed modules G are Warfield if and only if p s G is completely decomposable and Gap s G is Warfield for s bigger or equal to w. This is a strengthening of a result of Hill and Megibben [3]. Considering only the ordinal w we obtain a stronger result, namely, a module G is Warfield if and only if its first Ulm submodule p w G is completely decomposable and its zeroth Ulm factor Gap w G is simply presented. We show that for modules G with torsion zeroth Ulm factor the last result can be extended to ordinals s b w. However this breaks down if the module contains indicators of both the finite and w-type. Indeed we explicitly describe such a module G of torsion-free rank two which is Warfield but where Gap w1 G is not simply presented.

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“…[6,7,8,9]. As in [6] we denote by h 1 the subclass of h consisting of the modules of torsion-free rank one.…”

Section: Preliminariesmentioning

confidence: 99%

Section: Preliminariesmentioning

confidence: 99%

Characterizing a class of Warfield modules by Ulm submodules and Ulm factors

2001

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“…Let us quickly recall the main definitions and conventions. We adopt the description of mixed modules G in the class 7-/by generators and relations introduced in [6]. Let G be such a mixed module of torsion-free rank d with torsion submodule t G A basic generating system intrinsically defines a series of equations with coefficients in R that describe the relations among the generators.…”

Section: Preliminariesmentioning

confidence: 99%

“…By [6] all relation arrays are realized, i.e. there is a module in 7-/having a basic generating system with a prescribed relation array.…”

Section: Preliminariesmentioning

confidence: 99%

“…Boyer and Mader [2]. But here we prefer a representation of modules as extensions of torsion by torsionfree and denote the relations by the method established in [6], since there seems to be the possibility to extend the results to the bigger class of all modules of finite quotient p-rank, cf. [7], using a description of the torsion submodule by a straight basis, introduced by Benabdallah and Honda [1].…”

Section: Introductionmentioning

confidence: 99%

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Classification of mixed modules

Mutzbauer

Toubassi

1996

Acta Math Hung

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Characterizing a class of simply presented modules by relation arrays

1998

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Simply presented modules and Warfield modules are described in a class of mixed modules h with the property that the torsion submodule is a direct sum of cyclics and the quotient modulo the torsion submodule is divisible of arbitrary rank. Analogous to a result of Warfield it is shown that the mixed modules of torsion-free rank one are in some sense the building blocks of such modules. The results extend our previous work describing this class by relation arrays which are a natural outgrowth of their basic generating systems. Moreover, an intimate connection is shown between relation arrays and the indicators of modules. Furthermore, we prove two realizability results one of which is analogous to a theorem of Megibben for mixed modules of torsion-free rank one. One gives necessary and sufficient conditions on when a relation array of a module can realize an indicator of finite-type while the other shows that an admissable indicator can be realized by a simply presented module of torsion-free rank one. 0. Introduction. This paper characterizes the simply presented modules in a class of mixed modules h with the property that the torsion submodule is a direct sum of cyclics and the quotient modulo the torsion submodule is divisible of arbitrary rank. This class was described by generators and relations and the concept of a relation array and its properties studied in a series of papers [4,6,7,8]. In [6] we gave a splitting criterion in terms of the relation array which was extended in [7] to a larger class where the quotient p-rank is finite. In this paper we show the connection between the calculation of indicators done in [4] and the relation arrays. We then use this to describe the simply presented and Warfield modules in the class h.

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A Splitting Criterion for a Class of Mixed Modules

Cited by 9 publications

References 3 publications

Characterizing a class of Warfield modules by Ulm submodules and Ulm factors

Characterizing a class of Warfield modules by Ulm submodules and Ulm factors

Classification of mixed modules

Characterizing a class of simply presented modules by relation arrays

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