Reduced fuzzy primary decompositions for fuzzy ideals (Q2731732)

The author gives several characterizations for reduced fuzzy primary decompositions. These characterizations are used to obtain upper bounds for the number of components in such decompositions. An upper bound for the number of primary components in fuzzy primary decompositions for fuzzy ideals was given previously by the author and the reviewer. In this paper, new upper bounds are given and examples are provided showing that these bounds are sharp. Let \(R\) be a commutative ring with identity. Suppose that \(R\) has a composition series of length \(n\). Let \(\mu\) be a fuzzy ideal of \(R\) such that \(\text{Im} (\mu)= m\). Then the author shows that \(\mu\) has a fuzzy primary representation with no more than \((m-1)(2n+2-m)/2\) primary components. As a corollary, it is shown that if \(Q_1\cap Q_2 \cap\dots \cap Q_p\) is a reduced primary decomposition of \(\mu\), then \(p\leq n(n+1)/2\).