Related papers: A Dedekind Domain with Nontrivial Class Group

In this paper, as an extension of the integer case, we define polynomial functions over the residue class rings of Dedekind domains, and then we give canonical representations and counting formulas for such polynomial functions. In…

We present a generalization of a polynomial factorization algorithm that works with ideals in maximal orders of global function fields. The method presented in this paper is intrinsic in the sense that it does not depend on the embedding of…

Dedekind domains and their class groups are notions in commutative algebra that are essential in algebraic number theory. We formalized these structures and several fundamental properties, including number theoretic finiteness results for…

The theory of slice regular (also called hyperholomorphic) functions is a generalization of complex analysis originally given in the quaternionic framework, and then further extended to Clifford algebras, octonions, and to real alternative…

A commutative ring $R$ is stable if every non-zero ideal $I$ of $R$ is projective over its ring of endomorphisms. Motivated by a paper of Bass in the 1960s, stable rings have received wide attention in the literature ever since then. Much…

Let $R$ be an integral domain with $qf(R)=K$ and let $F(R)$ be the set of nonzero fractional ideals of $R.$ Call $R$ a dually compact domain (DCD) if for each $I\in F(R)$ the ideal $I_{v}=(I^{-1})^{-1}$ is a finite intersection of principal…

We consider the lattice-ordered groups Inv$(R)$ and Div$(R)$ of invertible and divisorial fractional ideals of a completely integrally closed Pr\"ufer domain. We prove that Div$(R)$ is the completion of the group Inv$(R)$, and we show there…

We generalize the theory of radical factorization from almost Dedekind domain to strongly discrete Pr\"ufer domains; we show that, for a fixed subset $X$ of maximal ideals, the finitely generated ideals with $\mathcal{V}(I)\subseteq X$ have…

We generalize the usual relationship between irreducible Zariski closed subsets of the affine space, their defining ideals, coordinate rings, and function fields, to a non-commutative setting, where "varieties" carry a PGL_n-action, regular…

$\DeclareMathOperator{\Int}{Int}\DeclareMathOperator{\IntR}{Int{}^\text{R}}$For a domain $D$, the ring $\Int(D)$ of integer-valued polynomials over $D$ is atomic if $D$ satisfies the ascending chain condition on principal ideals. However,…

Let L be the Leavitt path algebra of an arbitrary directed graph E over a field K. This survey article describes how this highly non-commutative ring L shares a number of the characterizing properties of a Dedekind domain or a Pr\"ufer…

The main result of this paper is a generalization of the theorem of Chevalley-Shephard-Todd to the rings of invariants of pseudoreflection groups over Dedekind domains. In the special case of a principal ideal domain in which the group…

We study different form of boundness for ideals of almost Dedekind domains, generalizing the notions of critical ideals, radical factorization, and SP-domains. We show that every almost Dedekind domain has at least one noncritical maximal…

Let $R$ be a normal Noetherian local domain of Krull dimension two. We examine intersections of rank one discrete valuation rings that birationally dominate $R$. We restrict to the class of prime divisors that dominate $R$ and show that if…

Let $D$ be a Dedekind domain with infinitely many maximal ideals, all of finite index, and $K$ its quotient field. Let $\operatorname{Int}(D) = \{f\in K[x] \mid f(D) \subseteq D\}$ be the ring of integer-valued polynomials on $D$. Given any…

The equivalence of the Kohn finite ideal type and the D'Angelo finite type with the subellipticity of the $\bar\partial$-Neumann problem is extended to pseudoconvex domains in $C^n$ whose defining function is in a Denjoy-Carleman…

Let $T''(X)$ and $T'(X)$ denote the collections of all real-valued functions on $X$ which are continuous on a dense cozero set and on an open dense subset of $X$ respectively. $T''(X)$ contains $C(X)$ and forms a subring of $T'(X)$ under…

We survey a few classes of analytic functions on the disk that have real boundary values almost everywhere on the unit circle. We explore some of their properties, various decompositions, and some connections these functions make to…

We have introduced and studied in [3] the class of Globalized multiplicatively pinched-Dedekind domains (GMPD domains). This class of domains could be characterized by a certain factorization property of the non-invertible ideals, (see [3,…

It is often stated that the Carlitz module is to the ring of univariate polynomials over a finite field what the multiplicative group is to the ring of integers. This analogy extends to the "rank 2" case, where Drinfeld modules play a role…