In this paper we provide explicit formulas for the core of an ideal. Recall that for an ideal I in a Noetherian ring R, the core of I, core(I), is the intersection of all reductions of I. For a subideal J is contained in I we say that J is a reduction of I, or that I is integral over J, if I~(r+1) = JI~r for some r ≥ 0; the smallest such r is called the reduction number of I with respect to J and is denoted by r_J(I). If (R, m) is local with infinite residue field k then every ideal has a minimal reduction, which is a reduction minimal with respect to inclusion.
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