2016/07/18 by Youngsu Kim, Kim, Youngsu, Louis J. Ratliff +3
Mathematics · #13 #Algebraic Geometry and Number Theory #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics #Rings, Modules, and Algebras
paper · pdf · doi:10.48550/arxiv.1607.05341
openalex publication_date 2016/07/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let I be a regular proper ideal in a Noetherian ring R, let e ≥ 2 be an integer, let \mathbf Te = R[u,tI,u(1)/(e)]' ∩ R[u(1)/(e),t(1)/(e)] (where t is an indeterminate and u =(1)/(t)), and let \mathbf re = u(1)/(e) \mathbf Te. Then the Itoh (e)-valuation rings of I are the rings (\mathbf Te/z)(p/z), where p varies over the (height one) associated prime ideals of \mathbf re and z is the (unique) minimal prime ideal in \mathbf Te that is contained in p. We show, among other things: (1) \mathbf re is a radical ideal if and only if e is a common multiple of the 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 = Qe, 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 = Qe and k = qe is a multiple of e, then there exists a unit θe ∈ V^* such that V^* = W[θe,u(1)/(k)] is a finite free integral extension domain of W, QV^* = N^*q, N^* = u(1)/(k)V^*, and [V^*:W] = k. (4) If the Rees integers of I are all equal to e, then V^* = W[θe] is a simple free integral extension domain of W, QV^* = N^* = u(1)/(e)V^*, and [V^*:W] = e = [(V^*/N^*):(W/Q)].