In the second part of the paper, we study torsion-free modules and mixed modules. We analyze the possibility of the isomorphism of two modules with isomorphic endomorphism rings. We touch on several questions about transitive and fully transitive modules.

Modules Over Discrete Valuation Domains

Classification theory of abelian groups, II: Local theory

By giving necessary and sufficient conditions for two isotype submodules of a p-local balanced projective group to be equivalent, we are able to introduce a general theory of isotype submodules of p-local balanced projective groups (or IB modules). Numerous applications of the above result are available particularly for the special class of IB modules introduced by Wick (known as SKT modules). We first show that the class of SKT modules is closed under direct summands, and then we are able to show that if H appears as an isotype submodule of the p-local balanced projective group G such that G/H is the coproduct of countably generated torsion groups, then H is an SKT module. Finally we show that IB modules satisfy general structural properties such as transitivity, full transitivity, and the equivalence of p'-high

/pdf/isotype-submodules-of-p-local-balanced-projective-groups-3hlap0ia4j.pdf

Isotype submodules of $p$-local balanced projective groups

On the structure of p-local abelian groups

In this paper a class of Abelian groups which includes the torsion totally projective groups, S-groups, and balanced projectives is studied. It is shown that this class of groups has a complete set of invariants. If p is a prime number then the ring of all rational numbers a/b with b relatively prime top will be denoted by Z . The category of ZP-modules are those Abelian groups with the property that multiplication by a prime other than the prime p is an automorphism of the group. If M is a Zp-module then (i) the torsion submodule of M is the maximal torsion subgroup of M; (ii) M is torsion-free if and only if the torsion submodule of M is (0); (iii) M is reduced if and only if M is a reduced group; and (iv) M is divisible if and only if M is a divisible group. Z and Q will denote the groups of integers and rational numbers respectively. The group Ext(Q/ZP, *) will be denoted by c(*). The limit ordinal X is a limited ordinal cofinal with w if there is a sequence of smaller ordinals /A, such that X = sup A3i Otherwise X is said to be not cofinal with

/pdf/a-classification-theorem-for-skt-modules-1ljjindbg0.pdf

A classification theorem for SKT-modules

In this paper a class of abelian groups (SKT-modules), which includes the torsion totally projective groups, S-groups, and balanced projectives is shown to be a subclass of a projective class of groups with respect to a naturally defined class of short exact sequences called the ch-projective modules and ch-pure sequences, respectively. Every ZP-module has a ch-pure projective resolution and every reduced ch-projective module is a summand of a SKT-module. It is finally shown that a ZP-module M is ch-projective if and only if, for every ordinal a, the two ZP-modules paM and M/p 'M are both ch-projective. 1. Definitions. If p is a prime number then the ring of all rational numbers a/b with b relatively prime top will be denoted by Z. The category of Z4-modules are those abelian groups with the property that multiplication by a prime other than the prime p is an automorphism of the group. If M is a Zp-module then (i) the torsion submodule of M is the maximal torsion subgroup of M; (ii) M is torsion-free if and only if the torsion submodule of M is (0); (iii) M is reduced if and only if M is a reduced group; and (iv) M is divisible if and only if M is a divisible group. Z and Q will denote the groups of integers and rational numbers respectively. The group Ext(Q/Zp, *) will be denoted by c(*). The limit ordinal X is a limit ordinal cofinal with w if there is a sequence of smaller ordinals fAi such that X = sup A.f Otherwise X is said to be not cofinal with

/pdf/a-projective-characterization-for-skt-modules-4hi0xtr38v.pdf

A projective characterization for SKT-modules

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A projective characterization for SKT-modules