Effective generators for fuzzy ideals (Q2709944)

Let \(R\) be a commutative ring. Suppose that \(R\) has a composition series of length \(n\). Then it is known that a fuzzy ideal \(\mu\) has a system of fuzzy generators consisting of no more than \(n^2-1\) elements. In this paper, the author improves this bound. Suppose that the image of \(\mu\) has cardinality \(m\). Then the author shows that \(\mu\) needs no more than \(m(2n+1-m) /2\) generators. Hence a fuzzy ideal of \(R\) has a system of fuzzy generators with no more than \(n(n-1)/2\) elements. If \(R\) is Artinian, then it is shown that every fuzzy ideal of \(R\) has a system of fuzzy generators with no more than \(1+n (n-1)/2\) elements. The author provides examples showing that these bounds are sharp.