Vol. 45, No. 2, 1973

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A class of divisible modules

Mark Lawrence Teply

Vol. 45 (1973), No. 2, 653–668

The 𝒯 -divisible R-modules are defined in terms of a hereditary torsion theory of modules over an associative ring R with identity element. In the special case where 𝒯 is the usual torsion class of modules over a commutative integral domain, the class of 𝒯 -divisible modules is precisely the class of divisible modules M such that every nonzero homomorphic image of M has a nonzero h-divisible submodule. In general, if 𝒯 is a stable hereditary torsion class, the class of Z-divisible modules satisfies many of the traditional properties of divisible modules over a commutative integral domain. This is especially true when F is Goldie’s torsion class 𝒢. For suitable 𝒯 the splitting of all 𝒯 -divisible modules is equivalent to h.d. Q𝒯1, where Q𝒯 is the ring of quotients naturally associated with 𝒯 Generalizations of Dedekind domains are studied in terms of 𝒯 -divisibility.

Mathematical Subject Classification
Primary: 16A21
Received: 28 February 1972
Published: 1 April 1973
Mark Lawrence Teply