Let (R,m) be a local
Cohen-Macaulay ring of dimension d and multiplicity e(R) = e. A natural question to
ask about an m-primary ideal I is whether there is any relation between the number
of generators of I and the least power t of m contained in I. (t will be called the
nilpotency degree of R∕I) It is quite straight forward to obtain a bound for v(I), the
number of generators in a minimal basis of I, in terms of t and e. However, there are
several interesting applications. The first is the existence of a bound for the
number of generators of any Cohen-Macaulay ideal I, i.e. any ideal I such that
R∕I is Cohen-Macaulay, in terms of e(R∕I), e(R) and height I. The second
application is a bound in terms of d and e for the reduction exponent of
m.
