A note on Itoh (e)-valuation rings of an ideal

作者：Kim Youngsu^*; Ratliff Louis J Jr; Rush David E

来源：Journal of Algebra, 2018, 498: 47-70.

DOI：10.1016/j.jalgebra.2017.11.008

• Commutative algebra
• Integral closures
• Rees valuations
• Rees integers
• Projective equivalence

Let I be a regular proper ideal in a Noetheriani ring R, let e >= 2 be an integer, let T-e = R[u, tI, u(1/e)]' boolean AND 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 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 = 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 theta(6) is an element of V* such that V* = W[theta(e),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[theta(e)] is a simple free integral extension domain of W, QV* = N* = u(1/e) V*, and [V* : W] = e = [(V*/N*) : (W/Q)].

• 出版日期2018-3-15