Simply Presented Tag Modules

A tag module is a generalization, in any abelian category, of a torsion abelian group. The theory of such modules is developed, it is shown that countably generated tag modules are simply presented, and that Ulm's theorem holds for simply presented tag modules. Zippin's theorem is stated and proved for countably generated tag modules. 1. TAG-modules In the theory of torsion abelian groups, a distinguished role is played by the cyclic groups of prime power order. Such a group is a uniserial module, by which we will mean a module whose submodules form a nite chain. The following two conditions on a module M were introduced by Singh in [14]. (I) Finitely generated submodules of homomorphic images of M are direct sums of uniserials. (II) If u and v are uniserial submodules of homomorphic images of M , and the length of u is no greater than the length of v, then any one-to-one map from a nonzero submodule of u to v can be extended to an isomorphism of u with a submodule of v. Singh's condition (II) was slightly di erent from this one but equivalent to it. Modules satisfying these two conditions were called S2-modules by Kahn [7] and TAGmodules by Benabdallah and Singh [3]. The idea is that a TAG-module is like a torsion abelian group. Certainly each torsion abelian group is a TAG-module, the uniserial submodules being cyclic groups of prime power order. Rather than working in a category of modules over a ring, we will work in an abelian category whose objects we will call modules. Normally we will want this category to have in nite direct sums and products, and to satisfy Grothendieck's axiom AB-5 (see [12]) on directed families Bi of submodules: A \ X Bi = X A \ Bi: It will also be convenient to have a distinguished monic S !M for each submodule of M so that submodules may be canonically considered as objects in the category.