NOTE ON GOOD IDEALS IN GORENSTEIN LOCAL RINGS

Abstract

Let I be an ideal in a Gorenstein local ring A with the maximal ideal m and d = dim A. Then we say that I is a good ideal in A, if I contains a reduction <TEX>$Q=(a_1,a_2,...,a_d)$</TEX> generated by d elements in A and <TEX>$G(I)=\bigoplus_{n\geq0}I^n/I^{n+1}$</TEX> of I is a Gorenstein ring with a(G(I)) = 1-d, where a(G(I)) denotes the a-invariant of G(I). Let S = A[Q/a<TEX>$_1$</TEX>] and P = mS. In this paper, we show that the following conditions are equivalent. (1) <TEX>$I^2$</TEX> = QI and I = Q:I. (2) <TEX>$I^2S$</TEX> = <TEX>$a_1$</TEX>IS and IS = <TEX>$a_1$</TEX>S：sIS. (3) <TEX>$I^2$</TEX>Sp = <TEX>$a_1$</TEX>ISp and ISp = <TEX>$a_1$</TEX>Sp ：sp ISp. We denote by <TEX>$X_A(Q)$</TEX> the set of good ideals I in <TEX>$X_A(Q)$</TEX> such that I contains Q as a reduction. As a Corollary of this result, we show that <TEX>$I\inX_A(Q)\Leftrightarrow\IS_P\inX_{SP}(Qp)$</TEX>.