Abstract
Let R be a commutative, semilocal, Noetherian domain, not a field. We say that R has finite representation type provided R has, up to isomorphism, only finitely many indecomposable finitely generated torsion-free modules. A special case (0.6) of our main theorem states that R has finite representation type if and only if
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1
R has Krull dimension 1;
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2
The integral closure \( \tilde R \) of R in its quotient field can be generated by 3 elements as an R-module; and
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3
The intersection of the maximal R-submodules of \( \tilde R/R \) is a cyclic R-module.
While there is no uniform bound on the number of indecomposable finitely generated torsion-free R-modules (as R varies among semilocal domains of finite representation type), the rank of every indecomposable is 1, 2, 3, 4, 5, 6, 8, 9 or 12. Moreover, there exists a semilocal domain R of finite representation type which has indecomposables of each of these ranks.
Cimen’ research was supported by a fellowship from Turkish government.
Both R. and S. Wiegand were partially supported by NSF grants while this research was being carried out.
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Cimen, N., Wiegand, R., Wiegand, S. (1995). One-Dimensional Rings of Finite Representation Type. In: Facchini, A., Menini, C. (eds) Abelian Groups and Modules. Mathematics and Its Applications, vol 343. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-0443-2_9
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