Integral closure and generalized transforms in graded domains

Graded Integral Domains and Nagata Rings , Ii

Let D be an integral domain with quotient field K, X be an indeterminate over D, K[X] be the polynomial ring over K, and R = {f ∈ K[X] | f(0) ∈ D}; so R is a subring of K[X] containing D[X]. For f = a0 + a1X + · · ·+ anX ∈ R, let C(f) be the ideal of R generated by a0, a1X, . . . , anX n and N(H) = {g ∈ R | C(g)v = R}. In this paper, we study two rings RN(H) and Kr(R, v) = { fg | f, g ∈ R, g 6=...

Kaplansky-type Theorems in Graded Integral Domains

It is well known that an integral domain D is a UFD if and only if every nonzero prime ideal of D contains a nonzero principal prime. This is the so-called Kaplansky’s theorem. In this paper, we give this type of characterizations of a graded PvMD (resp., G-GCD domain, GCD domain, Bézout domain, valuation domain, Krull domain, π-domain).

Integral Transforms of Fourier Cosine and Sine Generalized Convolution Type

Ratliff-rush Closure of Ideals in Integral Domains

This paper studies the Ratliff-Rush closure of ideals in integral domains. By definition, the Ratliff-Rush closure of an ideal I of a domain R is the ideal given by Ĩ := S (I :R I ) and an ideal I is said to be a Ratliff-Rush ideal if Ĩ = I. We completely characterize integrally closed domains in which every ideal is a Ratliff-Rush ideal and we give a complete description of the Ratliff-Rush cl...

On Complete Integral Closure and Archimedean Valuation Domains

Suppose D is an integral domain with quotient eld K and that L is an extension eld of K. We show in Theorem 4 that if the complete integral closure of D is an intersection of Archimedean valuation domains on K, then the complete integral closure of D in L is an intersection of Archimedean valuation domains on L; this answers a question raised by All rings considered in this paper are assumed to...