A note on Itoh (e)-Valuation Rings of and Ideal [article]

Youngsu Kim, Louis J. Ratliff, David E. Rush
2016 arXiv   pre-print
Let I be a regular proper ideal in a Noetherian ring R, let e > 2 be an integer, let T_e = R[u,tI,u^1/e]' ∩ R[u^1/e,t^1/e] (where t is an indeterminate and u =1/t), and let r_e = u^1/e T_e. Then the Itoh (e)-valuation rings of I are the rings ( T_e/z)_(p/z), where p varies over the (height one) associated prime ideals of r_e and z is the (unique) minimal prime ideal in T_e that is contained in p. We show, among other things: (1) r_e is a radical ideal if and only if e is a common multiple of
more » ... Rees integers of I. (2) For each integer k > 2, there is a one-to-one correspondence between the Itoh (k)-valuation rings (V^*,N^*) of I and the Rees valuation rings (W,Q) of uR[u,tI]; namely, if F(u) is the quotient field of W, then V^* is the integral closure of W in F(u^1/k). (3) For each integer k > 2, if (V^*,N^*) and (W,Q) are corresponding valuation rings, as in (2), then V^* is a finite integral extension domain of W, and W and V^* satisfy the Fundamental Equality with no splitting. Also, if uW = Q^e, and if the greatest common divisor of e and k is d, and c is the integer such that cd = k, then QV^* = N^*^c and [(V^*/N^*):(W/Q)] = d. Further, if uW = Q^e and k = qe is a multiple of e, then there exists a unit θ_e∈ V^* such that V^* = W[θ_e,u^1/k] is a finite free integral extension domain of W, QV^* = N^*^q, N^* = u^1/kV^*, and [V^*:W] = k. (4) If the Rees integers of I are all equal to e, then V^* = W[θ_e] is a simple free integral extension domain of W, QV^* = N^* = u^1/eV^*, and [V^*:W] = e = [(V^*/N^*):(W/Q)].
arXiv:1607.05341v1 fatcat:ziswfv4g5vff7nehuvgyymcuq4