ar X iv : m at h / 05 10 06 8 v 1 [ m at h . R A ] 4 O ct 2 00 5 INDECOMPOSABLE MODULES AND GELFAND RINGS

It is proved that a commutative ring is clean if and only if it is Gelfand with a totally disconnected maximal spectrum. It is shown that each indecomposable module over a commutative ring R satisfies a finite condition if and only if R P is an artinian valuation ring for each maximal prime ideal P. Commutative rings for which each indecomposable module has a local endomorphism ring are studied. These rings are clean and elementary divisor rings. It is shown that each commutative ring R with a Hausdorff and totally disconnected maximal spectrum is local-global. Moreover, if R is arithmetic then R is an elementary divisor ring. In this paper R is a commutative ring with unity and modules are unitary. In [11, Proposition 2] Goodearl and Warfield proved that each zero-dimensional ring R satisfies the second condition of our Theorem 1.1, and this condition plays a crucial role in their paper. In Section 1, we show that a ring R enjoys this condition if and only if it is clean, if and only if it is Gelfand with a totally disconnected maximal spectrum. So we get a generalization of results obtained by Anderson and Camillo in [1] and by Samei in [21]. We deduce that every commutative ring R with a Hausdorff and totally disconnected maximum prime spectrum is local-global, and moreover, R is an elementary divisor ring if, in addition, R is arithmetic. One can see in [7] that local-global rings have very interesting properties. In Section 3 we give a characterization of commutative rings for which each indecomposable module satisfies a finite condition: finitely generated, finitely co-generated, cyclic, cocyclic, artinian, noetherian or of finite length. We deduce that a commutative ring is Von Neumann regular if and only if each indecomposable module is simple. This last result was already proved in [5]. We study commuta-tive rings for which each indecomposable module has a local endomorphism ring. These rings are clean and elementary divisor rings. It remains to find valuation rings satisfying this property to give a complete characterization of these rings. We also give characterizations of Gelfand rings and clean rings by using properties of in-decomposable modules. Similar results are obtained in Section 4, for commutative rings for which each prime ideal contains only one minimal prime ideal. We denote respectively Spec R, Max R and Min R, the space of prime ideals, maximal ideals, and minimal …