Decomposing torsion modules |
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Authors: | Willy Brandal Erol Barbut |
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Affiliation: | Department of Mathematics and Applied Statistics , University of Idaho , Moscow, Idaho, 83843 |
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Abstract: | E. Matlis proved that if R is an integral domain with quotient field Q and K is the R-module Q/R, then all torsion R-modules decompose into a direct sum of local submodules if and only if K decomposes into a direct sum of local submodules. Thus K is a test module to determine whether torsion modules decompose. We generalize this result to commutative rings. If R is a commutative ring and a torsion theory of R is given by a Gabriel topology , then form the ring of quotients R and let K be the cokernel of the canonical ring homomorphism from R to R. In some special cases, every -torsion R-module decomposes into a direct sum of local submodules if and only if K decomposes. However, there is an example where this is not the case. The principal result is: given R, and K, there is a related filter K of ideals of R, which is a subset of , such that all K-pretorsion R-modules decompose into a direct sum of local submodules if and only if K decomposes. The relationship between and K is investigated. |
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