Finite Rank Torsion Research Articles

The Grothendieck and Picard groups of finite rank torsion free mathfrak {sl}(2)-modules

The classification problem for simple {mathfrak {sl}(2)}-modules leads in a natural way to the study of the category of finite rank torsion free {mathfrak {sl}(2)}-modules and its subcategory of rational {mathfrak {sl}(2)}-modules. We prove that the rationalization functor induces an identification between the isomorphism classes of simple modules of these categories. This raises the question of what is the precise relationship between other invariants associated with them. We give a complete solution to this problem for the Grothendieck and Picard groups, obtaining along the way several new results regarding these categories that are interesting in their own right.

Completely decomposable direct summands of torsion-free abelian groups of finite rank

Let $A$ be a finite rank torsion--free abelian group. Then there exist direct decompositions $A=B\oplus C$ where $B$ is completely decomposable and $C$ has no rank 1 direct summand. In such a decomposition $B$ is unique up to isomorphism and $C$ unique up to near-isomorphism.

一类有限秩Abel群的自同构

本文给出了一类有限秩的具有C2自同构的无挠Abel群。 In this paper, we mainly study a class of finite rank torsion free abelian groups which their automorphism groups are C2.

Quasi-decompositions for Self-Small Abelian Groups#

We will prove a Krull-Schmidt type theorem for the quasi decompositions of a self-small abelian group of finite torsion free rank, which extends the classical result proved for finite rank torsion free abelian groups.

The integral closure of [formula omitted] in the Grothendieck ring for quasi-isomorphism classes of finite rank torsion free modules over a Dedekind domain

The integral closure of [formula omitted] in the Grothendieck ring for quasi-isomorphism classes of finite rank torsion free modules over a Dedekind domain

Grothendieck rings for certain categories of quasihomomorphisms of torsion free modules over Dedekind domains

In this paper we consider the Krull-Schmidt-Grothendieck ring K(Q) for certain full subcategories Q of the category of quasihomomorphisms of finite rank torsion free modules over a Dedekind domain W. More specifically, we will be concerned with modules G such that W’ @ G is a Butler W’-module for some suitable finite integral extension W’ of W. We will usually also ‘assume that the typeset of W’ @ G consists of idempotent types. Let K = K(@‘) and let N = nil rad K. By a strongly indecomposable domain in Q, we mean a W-algebra D which is a commutative integral domain and such that the underlying W-module of D is strongly indecomposable and belongs to Q. If D is such a domain and G is any torsion free W-module, we let D-rank G = rank, Hom(G, D). We set P, = {[Cl [H] E K ] D-rank G = D-rank H}. We let E be the subring of K generated by all [D] such that D is a strongly indecomposable domain in 5F. Consider the following four properties:

Strongly homogeneous torsion free abelian groups of finite rank

An abelian group is strongly homogeneous if for any two pure rank 1 subgroups there is an automorphism sending one onto the other. Finite rank torsion free strongly homogeneous groups are characterized as the tensor product of certain subrings of algebraic number fields with finite direct sums of isomorphic subgroups of Q Q , the additive group of rationals. If G G is a finite direct sum of finite rank torsion free strongly homogeneous groups, then any two decompositions of G G into a direct sum of indecomposable subgroups are equivalent.

Endomorphism Rings and Direct Sums of Torsion Free Abelian Groups

Properties of abelian groups related to a given finite rank torsion free abelian group A are analyzed in terms of End (A), the endomorphism ring of A. This point of view gives rise to generalizations of some classical theorems by R. Baer and examples of pathological direct sum decompositions of finite rank torsion free abelian groups.

Nearly isomorphic torsion free abelian groups

Let K be the Krull-Schmidt-Grothendieck group for the category of finite rank torsion free abelian groups. The torsion subgroup T of K is determined and it is proved that K T is free. The investigation of T leads to the concept of near isomorphism, a new equivalence relation for finite rank torsion free abelian groups which is stronger than quasiisomorphism.

Endomorphism rings and direct sums of torsion free abelian groups

Properties of abelian groups related to a given finite rank torsion free abelian group A are analyzed in terms of End (A), the endomorphism ring of A. This point of view gives rise to generalizations of some classical theorems by R. Baer and examples of pathological direct sum decompositions of finite rank torsion free abelian groups.

Summands of finite rank torsion free abelian groups

A finite rank torsion free abelian group has, up to isomorphism, only finitely many summands.

Almost Completely Decomposable Torsion Free Abelian Groups

A finite rank torsion free abelian group $G$ is almost completely decomposable if there exists a completely decomposable subgroup $C$ with finite index in $G$. The minimum of $[G:C]$ over all completely decomposable subgroups $C$ of $G$ is denoted by $i(G)$. An almost completely decomposable group $G$ has, up to isomorphism, only finitely many summands. If $i(G)$ is a prime power, then the rank 1 summands in any decomposition of $G$ as a direct sum of indecomposable groups are uniquely determined. If $G$ and $H$ are almost completely decomposable groups, then the following statements are equivalent: (i) $G \oplus L \approx H \oplus L$ for some finite rank torsion free abelian group $L$. (ii) $i(G) = i(H)$ and $H$ contains a subgroup $G’$ isomorphic to $G$ such that $[H:G’]$ is finite and prime to $i(G)$. (iii) $G \oplus L \approx H \oplus L$ where $L$ is isomorphic to a completely decomposable subgroup with finite index in $G$.

Almost completely decomposable torsion free abelian groups

A finite rank torsion free abelian group G G is almost completely decomposable if there exists a completely decomposable subgroup C C with finite index in G G . The minimum of [ G : C ] [G:C] over all completely decomposable subgroups C C of G G is denoted by i ( G ) i(G) . An almost completely decomposable group G G has, up to isomorphism, only finitely many summands. If i ( G ) i(G) is a prime power, then the rank 1 summands in any decomposition of G G as a direct sum of indecomposable groups are uniquely determined. If G G and H H are almost completely decomposable groups, then the following statements are equivalent: (i) G ⊕ L ≈ H ⊕ L G \oplus L \approx H \oplus L for some finite rank torsion free abelian group L L . (ii) i ( G ) = i ( H ) i(G) = i(H) and H H contains a subgroup G ′ G’ isomorphic to G G such that [ H : G ′ ] [H:G’] is finite and prime to i ( G ) i(G) . (iii) G ⊕ L ≈ H ⊕ L G \oplus L \approx H \oplus L where L L is isomorphic to a completely decomposable subgroup with finite index in G G .

A class of pure subgroups of completely decomposable abelian groups

The class M of pure subgroups of completely decomposable torsion free abelian groups of finite rank is considered by Butler [2]. Many of the examples of pathological direct sum decompositions of finite rank torsion free abelian groups occur in the class .4. On the other hand, homogeneous a-groups are completely decomposable and if A is an s-group, then typeset(A) is finite and closed under intersection of types. This note considers a-groups with typesets of cardinality at most 4. Motivation is provided by the curious results of Cruddis [3]. An easier and more illuminating proof of these results is a consequence of the following theorems. Let A= {-To, TI, T2, T73} be a set of types with To =rl1r -72r lT3; let .A be the class of a-groups with typeset c A; let :A be the sum of types in A; and let >,A be the sum of types in typeset(A). Define T to be the type represented by (Xo) and, forp a prime, let T-D be the type represented by (ne) where n2,=O and nQ= xo if p#q.