On the boundedness and the unboundedness of the number of generators of ideals and multiplicity

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For an unmixed ideal I in a regular formal power series ring R = k[[ x 1,…, x n ]] with the Krull dim of R I = 1 or dim R I = 2 or dim R I = 3 and R I integrally closed in its field of quotients, one can give an upper bound for the number of generators for I over R in terms of the multiplicity of R I and the embedding dimension of R I . For dim R I = 4 an example is given to show this is not possible, even for prime ideals.