We describe all isomorphism classes of finitely generated Λ-modules, where Λ is any complete local (commutative noetherian) ring whose category of modules of finite length does not have wild representation type. (There is a possible exception to our results, involving characteristic 2.)

In trying to extend the concept of torsion to rings more general than commutative integral domains the first thing that we notice is that if the definition is carried over word for word, integral domains are the only rings with torsion-free modules. Thus, if m is an element of any right module M over a ring containing a pair of non-zero elements x and y such that xy = 0, then either mx = 0 or (mx)y = 0. A second difficulty arises in the non-commutative case: Does the set of torsion elements of M form a submodule? The answer to this question will not even be "yes" for arbitrary non-commutative integral domains.

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This is the first of a series of four papers describing the finitely generated modules over all commutative noetherian rings that do not have wild representation type (with a possible exception involving characteristic 2). This first paper identifies the wild rings, in the complete local case. The second paper describes the finitely generated modules over the remaining complete local rings. The last two papers extend these results by dropping the "complete local" hypothesis.