Primary decompositions of fuzzy submodules
Let \(R\) be a commutative ring with identity and let \(M\) be a unitary module over \(R\). In this paper, the authors consider the decomposition of a fuzzy submodule of \(M\) into an intersection of primary fuzzy submodules of \(M\). The authors prove two uniqueness theorems concerning such a decomposition. They show that such a decomposition exists when \(M\) is Noetherian. They also consider the notions of reduced or irredundant primary decompositions, irreducible decompositions, isolated and embedded prime ideals. The authors provide some pertinent counterexamples. They apply their decomposition theory to characterize an infinite intersection of fuzzy submodules of a Noetherian module as a zero fuzzy submodule. The results are analogues of the Krull intersection theorem.
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