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# Ranks and bounds for indecomposable modules over one-dimensional Noetherian rings

#### Abstract

We consider one-dimensional, reduced Noetherian rings * R* with finite normalization. We assume that there exists a positive integer *N _{R}* such that, for every indecomposable finitely generated torsion-free

*R*-module

*M*and for every minimal prime ideal

*P*of

*R,*the dimension of

*M*, as a vector space over the field

_{P}*R*localized at

_{P}, R*P,*is less than or equal to

*N*. We call the set of vector-space dimensions the

_{R}*rank-set*of the module. We often call the sequence of vector-space dimensions the

*rank*of the module. We are interested in what rank-sets occur for indecomposable

*R*-modules. ^ In Chapter 2, with Meral Arnavut and Sylvia Wiegand, many possible ranks are eliminated. In the constant rank case, that is, when we have a module

*M*such that the vector-space dimension of

*M*is the same for every minimal prime

_{P}*P*of

*R,*the only ranks occurring for indecomposable modules are 1,2,3,4 and 6. In the non-constant rank case, we show that for

*n*≥ 8 there are no indecomposable modules with rank-sets between

*n*and 2

*n*− 8. On the other hand, for each

*n*≥ 8, we construct an indecomposable module with rank-set the set of consecutive integers from

*n*to 2

*n*− 7. ^ In Chapter 3, for each set of consecutive integers not ruled out in Chapter 2, we produce a semilocal ring and an indecomposable module over that ring having that set as its rank-set. Could other ranks occur for indecomposable modules? To answer this, we construct some indecomposable modules with ranks of non-consecutive integers. We then give some conditions to show some additional sets of non-consecutive integers never occur as the rank-sets of indecomposable modules. ^ In Chapter 4, with Nick Baeth, we prove the last case in a theorem listing the ranks that occur for indecomposable modules over a local ring. This result was previously known with an additional hypothesis on the characteristic of

*R*and its residue field. This hypothesis was assumed in the work of Chapter 2, but now we have eliminated it, and so, the results in Chapter 2 hold in more generality. ^

#### Subject Area

Mathematics

#### Recommended Citation

Luckas, Melissa R, "Ranks and bounds for indecomposable modules over one-dimensional Noetherian rings" (2007). *ETD collection for University of Nebraska - Lincoln*. AAI3283908.

http://digitalcommons.unl.edu/dissertations/AAI3283908