An invariant for modules over a discrete valuation ring

Stanton, Robert O.

doi:10.1090/s0002-9939-1975-0360572-8

Cited by 17 publications

(5 citation statements)

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“…Now fz + gz is admissible for each z E Z, so using a well-known We begin by extending to valuated groups an invariant introduced by Warfield [11] and generalized by Stanton [9]. Our approach parallels that of Stanton.…”

Section: Valuated Trees and Groupsmentioning

confidence: 99%

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Existence Theorems for Warfield Groups

Hunter¹,

Richman²,

Walker³

1978

Transactions of the American Mathematical Society

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Abstract. Warfield studied p-local groups that are summands of simply presented groups, introducing invariants that, together with Ulm invariants, determine these groups up to isomorphism. In this paper, necessary and sufficient conditions are given for the existence of a Warfield group with prescribed Ulm and Warfield invariants. It is shown that every Warfield group is the direct sum of a simply presented group and a group of countable torsion-free rank. Necessary and sufficient conditions are given for when a valuated tree can be embedded in a tree with prescribed relative Ulm invariants, and for when a valuated group in a certain class, including the simply presented valuated groups, admits a nice embedding in a countable group with prescribed relative Ulm invariants. These conditions, which are intimately connected with the existence of Warfield groups, are given in terms of new invariants for valuated groups, the derived Ulm invariants, which vanish on groups and fit into a six term exact sequence with the Ulm invariants.

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Section: Valuated Trees and Groupsmentioning

confidence: 99%

“…Next we show, by induction on n, that pHn(9) is nice for every 9. Since H"+X/Hn is finite, so are Hn+X(9)/Hn (9) and pHn+x(9)/pHn (9).…”

Section: Valuated Trees and Groupsmentioning

confidence: 99%

Existence Theorems for Warfield Groups

Hunter¹,

Richman²,

Walker³

1978

Transactions of the American Mathematical Society

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“…THEOREM 2. (Warfield (1975) In the above theorem, / represents the Ulm invariants while S represents the invariants in Stanton (1975).…”

Section: Introductionmentioning

confidence: 99%

Decompositions of modules over a discrete valuation ring

Stanton

1979

J. Aust. Math. Soc.

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“…He used these invariants to prove a generalized version of Ulm's theorem; namely, two Warfield groups are isomorphic if and only if they have the same Ulm and Warfield invariants. Stanton [6] later extended these invariants to all groups. An alternate development of Warfield's theory was given by Hunter, Richman and Walker in [3].…”

mentioning

confidence: 99%