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.)

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.

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Abstract. A commutative Noetherian local ring (R, m, k) is called Dedekindlike provided R is one-dimensional and reduced, the integral closure R is generated by at most 2 elements as an R-module, and m is the Jacobson radical of R. If M is an indecomposable finitely generated module over a Dedekind-like ring R, and if P is a minimal prime ideal of R, it follows from a classification theorem due to L. Klingler and L. Levy that M P must be free of rank 0, 1 or 2.Now suppose (R, m, k) is a one-dimensional Cohen-Macaulay local ring that is not Dedekind-like, and let P 1 , . . . , P t be the minimal prime ideals of R. The main theorem in the paper asserts that, for each non-zero t-tuple (n 1 , . . . , n t ) of non-negative integers, there is an infinite family of pairwise non-isomorphic indecomposable finitely generated R-modules M satisfyingIn 1911, E. Steinitz determined the structure of all finitely generated modules over Dedekind domains. This structure is so simple that one is tempted to try to generalize Steinitz's result to a larger class of commutative rings. Indeed, in a recent series of papers [KL1], [KL2], [KL3] L. Klingler and L. Levy presented a classification, up to isomorphism, of all finitely generated modules over a class of commutative rings they call "Dedekind-like".We recall that a commutative Noetherian local ring (R, m, k) is Dedekind-like [KL1, Definition 2.5] provided R is one-dimensional and reduced, the integral closure R is generated by at most 2 elements as an R-module, and m is the Jacobson radical of R. (In [KL2, (1.1.3)] a further requirement is imposed: If R/m is a field, then it is a separable extension of k. Klingler and Levy prove their classification theorem only under this additional hypothesis. In the present paper, however, we do not require that Dedekind-like rings satisfy this separability condition.) Although Dedekind-like rings are very close to their normalizations, their module structure is much more complicated than that of Dedekind domains. Klingler and Levy dash any hope of a further extension of their classification theorem by showing that, if R is not a homomorphic image of a Dedekind-like ring or a special type

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