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)]. (C) 2017 Elsevier Inc. All rights reserved.

The M-graded domains, which are almost Schreier are classified under the assumption that the integral closure of R is a root extension of R, where M is a torsion-free, commutative, cancellative monoid. In the case that D[M] is a commutative monoid domain it is shown that if M conical and is a root extension, then D[M] is almost Schreier if and only if M and D are almost Schreier. If R=Z[n] is an order in a quadratic extension field of , it is shown that the conditions; R[X] is IDPF; R[X] is inside factorial; R[X] is almost Schreier; is a root extension; and every prime divisor of n also divides the discriminant of the extension K/Q; are equivalent conditions.

Let R be a locally quasi-unmixed domain, a,b1,…,bn an asymptotic sequence in R, I=(a,b1,…,bn)R and S=R[b1/a,…,bn/a]=R[I/a], the monoidal transform of R with respect to I. It is shown that S is a locally quasi-unmixed domain, a,b1/a,…,bn/a is an asymptotic sequence in S and there is a one-to-one correspondence between the asymptotic primes Aˆ⁎(I) of I and the asymptotic primes Aˆ⁎(aS) of aS=IS. Moreover, if a,b1,…,bn is an R-sequence, this extends to a one-to-one correspondence between AssR(R/I) and AssS(S/aS). In the case that R is a unique factorization domain, the height one prime ideals of S are examined to determine how far S is from being a UFD. A complete description is given of which height one prime ideals P of S are principal or have a principal primary ideal in the case that P∩R has height 1. If the prime divisors of a satisfy a mild condition, a similar description is given in the case that P∩R has height >1. These are applied to give similar results for the Rees ring R[1/t,It] where t is an indeterminate.

It is shown that if S is a commutative multiplicative monoid with a zero element, the set C(S) of congruences on S is a lattice module over the multiplicative lattice L(S) of ideals of S. This gives a method of obtaining results on the set of congruences on a monoid by extending results on modules to lattice modules, and shows the relevance of examples and results on set of congruences on a monoid to the development of results on lattice modules. We illustrate these two aspects by extending specific results on modules to lattice modules, to get corresponding results on the set of congruences on a monoid, and by extending a well-known result of Drbohlav on the set of congruences on a monoid, to obtain a more general primary decomposition theorem on lattice modules than those in the literature.

The class of m-full and four related classes of ideals in a local ring (R, m) are extended by replacing m with other ideals and the resulting classes of ideals are compared. It is shown that contracted ideals are m-full in a local ring with infinite residue field.

Some characterizations are given of A. R. Naghipour's strongly prime submodules among the prime submodules of a finitely generated R-module M, together with some consequences of these including counterparts for modules to classical results on G-ideals and Hilbert rings.

Let R be an integral domain with quotient field K, and let Int(R) be the ring of integer-valued polynomials on R. That is Int(R) = {f ∈ K[X] | f(R) ⊆ R}. It is known that if R is an almost Dedekind domain, then the double boundedness condition, which characterizes when Int(R) is a Prüfer domain, is equivalent to another double boundedness condition which arises in characterizing when Int(R) ≠ R[X]. After recasting the first of these conditions so that it applies to any integral domain R, instead of just to Prüfer domains, it is shown that these two double boundedness conditions are equivalent if R is a finite conductor domain, but not for all integral domains R.

Computations are given of the resultants Res(s(m) , s(n)) of pairs of Selmer polynomials s(n) - s(n) (X) - X-n - X - 1. It is shown that for each fixed m is an element of N the sequence of integers Res(s(m) , s(m) broken vertical bar k) = Res(s(m)(X), X-k -1) satisfies a simple linear recursion which can be described in terms of higher Lucas sequences.

It is shown that the well-known characterizations of when a commutative ring R has Noetherian spectrum carry over to characterizations of when the set Spec(M) of prime submodules of a finitely generated module M is Noetherian. The symmetric algebra S-R(M) of M is used to show that the Noetherian property of Spec(R), and some related properties, pass from the ring R to the finitely generated R-modules.

Let V be a rank one discrete valuation ring (DVR) on a field F and let L/F be a finite separable algebraic field extension with [L:F] = m. The integral closure of V in L is a Dedekind domain that encodes the following invariants: (i) the number of extensions of V to a valuation ring W on L, (ii) the residue degree of each W over V, and (iii) the ramification degree of each W over V. Given a finite set of DVRs on F, an m-consistent system is a family of sets enumerating what is theoretically possible for the above invariants of each V in the set. The m-consistent system is realizable if there exists a finite separable extension field L/F that gives for each V the listed invariants. We investigate the realizability of m-consistent systems.

Let I be a regular proper ideal in a Noetherian ring R. We prove that there exists a simple free integral extension ring A of R such that the ideal IA has a Rees-good basis; that is, a basis c1,…,cg such that ciW=IW for i=1,…,g and for all Rees valuation rings W of IA. Moreover, A may be constructed so that: (i) IA and I have the same Rees integers (with possibly different cardinalities), and (ii) AP is unramified over RP∩R for each asymptotic prime divisor P of IA. Indeed, if H is a regular ideal in R such that each asymptotic prime divisor of H is contained in an asymptotic prime divisor of I, then (ii) holds for HA. If Card(ReesH)⩽Card(ReesI), we prove that (i) also holds for HA and H. If I=(b1,…,bg)R and b1,…,bg is an asymptotic sequence, we prove that b1,…,bg is a Rees-good basis of I.

Let I be a proper nonnilpotent ideal in a local (Noetherian) ring (R, M) and let be a reduction of I; that is, J subset of I and = J I(n) = I(n+1) for some nonnegative integer n. We prove that there exists a finite free local unramified extension ring S of R such that the ideal IS has a minimal reduction K subset of JS with the property that the number of elements in a minimal generating set of K is equal to the analytic spread of K and thus also equal to the analytic spread of I.

Contractedness of 𝔪-primary integrally closed ideals played a central role in the development of Zariski's theory of integrally closed ideals in two-dimensional regular local rings (R, 𝔪). In such rings, the contracted 𝔪-primary ideals are known to be characterized by the property that I: 𝔪 = I: x for some x ∈ 𝔪 ∖𝔪2. We call the ideals with this property full ideals and compare this class of ideals with the classes of 𝔪-full ideals, basically full ideals, and contracted ideals in higher dimensional regular local rings. The 𝔪-full ideals are easily seen to be full. In this article, we find a sufficient condition for a full ideal to be 𝔪-full. We also show the equivalence of the properties full, 𝔪-full, contracted, integrally closed, and normal, for the class of parameter ideals. We then find a sufficient condition for a basically full parameter ideal to be full.

Let I be a regular proper ideal in a Noetherian ring R, and let P(I) be the set of all integrally closed ideals J of R that are projectively equivalent to I. It is known that there is naturally associated to I a positive integer d and an additive subsemigroup {ci|i+} of + such that [image omitted], where [image omitted] is the integrally closed ideal {xR|xd is in the integral closure (Ici)a of Ici}. It is shown here that the sequences of ideals [image omitted] and [image omitted] are filtrations on R, and that we have subfiltrations {(Ii)a}i0=ff*e. Associated with these filtrations, we have graded rings [image omitted], and these inclusion maps induce isomorphisms on the homogeneous prime spectra of these graded rings. This raises the question of when the homogeneous prime spectrum HSpec(R[u,tf*]) of R[u,tf*] is isomorphic to either HSpec(R[u,tf]) or HSpec(R[u,te]). It is shown that HSpec(R[u,tf*]) HSpec(R[u,tf]) if and only if c1=1 if and only if P(I) is projectively full.

It is shown how to associate to any polytope that is not a simplex and any field K, a commutative integral domain D which has no irreducible elements and which is not pre-Schreier. The integral domain D is a generalized power series ring over K.

Let I be a nonzero proper ideal in a Noetherian integral domain R. In this paper we establish the existence of a finite separable integral extension domain A of R and a positive integer m such that all the Rees integers of IA are equal to m. Moreover, if R has altitude one, then all the Rees integers of J=Rad(IA) are equal to one and the ideals Jm and IA have the same integral closure. Thus Rad(IA)=J is a projectively full radical ideal that is projectively equivalent to IA. In particular, if R is Dedekind, then there exists a Dedekind domain A having the following properties: (i) A is a finite separable integral extension of R; and (ii) there exists a radical ideal J of A and a positive integer m such that IA=Jm. In this case the extension A also has the property that for each maximal ideal N of A with I⊆N, the canonical inclusion R/(N∩R)↪A/N is an isomorphism, and the integer m is a multiple of [A(0):R(0)].

It is shown that the complete integral closure of a seminormal Mori lattice is a Krull lattice. This extends a result of V. Barucci to multiplicative lattices. New results on rings are obtained by specializing to the case that the lattice is the set of homogeneous ideals of a graded ring R = circle plus(alpha m)R(alpha),, where M is an abelian cancellative torsion-free monoid.