Finite direct sums of cyclic valuated<i>p</i>-groups

Hunter, Roger; Richman, Fred; Walker, Elbert A.

doi:10.2140/pjm.1977.69.97

Cited by 13 publications

(12 citation statements)

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“…Recall from [HRW77] ℓ)) so f cannot be extended to End(G), a contradiction. (1) f can be extended to an endomorphism of G; (2) f can be extended to a homomorphism f : K → G; (3) f can be extended to a homomorphism f :…”

Section: Finite Subgroups In Q(g)mentioning

confidence: 94%

Subgroups which admit extensions of homomorphisms

2013

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Section: Finite Subgroups In Q(g)mentioning

confidence: 94%

Subgroups which admit extensions of homomorphisms

2013

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“…Because, pE(B) is a proper ideal of E(B), we have pB = 0. By [8], B is a cyclic group. Hence, A is a direct sum of cyclic groups of order p. R Φ R φ = φ.…”

Section: Hereditary and Quasi-frobenius Endomorphism Ringsmentioning

confidence: 99%

“…The valuated p-local groups are the objects of the category V p studied extensively by Hunter, Richman and Walker (e.g. see [7], [8] and [11]). A group homomorphism α : (G, v) → (H, w) is a V p -morphism if w(α(x)) ≥ v(x) for all x ∈ G, and we write α ∈ Mor(G, H) in this case.…”

Section: Introductionmentioning

confidence: 99%

Finite valuated groups as modules over their endomorphism ring

Albrecht

2017

Stud. Univ. Babes-Bolyai Math.

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Abstract. This paper discusses the structure of a finite valuated p-group when viewed as a module over its endomorphism ring. A category equivalence between full subcategories of the category of valuated p-groups and the category of right modules over the endomorphism ring of A is used to investigate the interaction between this module structure and homological properties of the underlying group. Examples are given throughout the paper.

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“…Since Λ is uniserial, each picket (B; A) is All pickets and all indecomposables of type Q t s have the property that the subgroup is either zero or cyclic. This is always the case for indecomposable objects in S 2 (Λ) according to [2,Theorem 4].…”

Section: History and Introductionmentioning

confidence: 99%

ABELIAN GROUPS WITH A p²-BOUNDED SUBGROUP, REVISITED

Petroro

Schmidmeier

2011

J. Algebra Appl.

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Let Λ be a commutative local uniserial ring of length n, p be a generator of the maximal ideal, and k be the radical factor field. The pairs (B, A) where B is a finitely generated Λ-module and A ⊆ B a submodule of B such that p m A = 0 form the objects in the category Sm(Λ). We show that in case m = 2 the categories Sm(Λ) are in fact quite similar to each other: If also ∆ is a commutative local uniserial ring of length n and with radical factor field k, then the categories S 2 (Λ)/N Λ and S 2 (∆)/N ∆ are equivalent for certain nilpotent categorical ideals N Λ and N ∆ . As an application, we recover the known classification of all pairs (B, A) where B is a finitely generated abelian group and A ⊆ B a subgroup of B which is p 2 -bounded for a given prime number p. determined uniquely by the lengths and m of the Λ-modules B and A; we write P m = (B; A).Clearly, the complexity of categories of type S m (Λ) increases with m. The categories S 0 (Λ) and mod Λ are equivalent so the only indecomposable objects in S 0 (Λ) are the pickets of type P 0 . In the category S 1 (Λ) (which we consider briefly in Sec. 3), every indecomposable object is a picket of type P 0 or P 1 . The category S 2 (Λ) contains additional indecomposables which are not pickets; it turns out that an invariant which has been introduced by Prüfer [3, §7] in 1923 provides an efficient classification. Definition. LetB be a Λ-module and a ∈ B be a nonzero element. The height exponent of a is h B (a) = h(a) = max{n ∈ N 0 : a = p n b for some b ∈ B}; the height sequence H B (a) = (h(a), h(pa), . . . , h(p a)) consists of the height exponents of the nonzero p-power multiples of a. Example 1.1. For pickets, the height sequence consists of consecutive numbers: In a picket (B; A) = P m where m > 0, any generator for A has the height sequence ( − m, − m + 1, . . . , − 1). J. Algebra Appl. 2011.10:377-389. Downloaded from www.worldscientific.com by UNIVERSITY OF AUCKLAND LIBRARY -SERIALS UNIT on 06/21/15. For personal use only.

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Finite direct sums of cyclic valuatedp-groups

Cited by 13 publications

References 1 publication

Subgroups which admit extensions of homomorphisms

Subgroups which admit extensions of homomorphisms

Finite valuated groups as modules over their endomorphism ring

ABELIAN GROUPS WITH A p²-BOUNDED SUBGROUP, REVISITED

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Finite direct sums of cyclic valuatedp-groups

Cited by 13 publications

References 1 publication

Subgroups which admit extensions of homomorphisms

Subgroups which admit extensions of homomorphisms

Finite valuated groups as modules over their endomorphism ring

ABELIAN GROUPS WITH A p2-BOUNDED SUBGROUP, REVISITED

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ABELIAN GROUPS WITH A p²-BOUNDED SUBGROUP, REVISITED