Principal Ideals Research Articles

Increasing sequences of principal left ideals of $\beta \mathbb{Z}$ are finite

We show that increasing sequences of principal left ideals of $\beta \mathbb {Z}$ are finite. As a consequence, $\beta \mathbb {Z}\setminus \mathbb {Z}$ is a disjoint union of maximal principal left ideals of $\beta \mathbb {Z}$. Another consequence is th

Editorial remarks on Y. Zelenyuk's proof of the nonexistence of infinite increasing chains of principal left ideals in $\beta \mathbb Z $

Editorial remarks on Y. Zelenyuk's proof of the nonexistence of infinite increasing chains of principal left ideals in $\beta \mathbb Z $

On the orbits of automaton semigroups and groups

We investigate the orbits of automaton semigroups and groups to obtain algorithmic and structural results, both for general automata but also for some special subclasses. First, we show that a more general version of the finiteness problem for automaton groups is undecidable. This problem is equivalent to the finiteness problem for left principal ideals in automaton semigroups generated by complete and reversible automata. Then, we look at w-word (i.\,e.\ right infinite words) with a finite orbit. We show that every automaton yielding an w-word with a finite orbit already yields an ultimately periodic one, which is not periodic in general, however. On the algorithmic side, we observe that it is not possible to decide whether a given periodic w-word has an infinite orbit and that we cannot check whether a given reversible and complete automaton admits an w-word with a finite orbit, a reciprocal problem to the finiteness problem for automaton semigroups in the reversible case. Finally, we look at automaton groups generated by reversible but not bi-reversible automata and show that many words have infinite orbits under the action of such automata.

Revisiting G-Dedekind domains

AbstractLet 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 fractional ideals. We characterize DCDs and show that the class of DCDs properly contains various classes of integral domains, such as Noetherian, Mori, and Krull domains. In addition, we show that a Schreier DCD is a greatest common divisor (GCD) (Greatest Common Divisor) domain with the property that, for each $A\in F(R)$ , the ideal $A_{v}$ is principal. We show that a domain R is G-Dedekind (i.e., has the property that $A_{v}$ is invertible for each $A\in F(R)$ ) if and only if R is a DCD satisfying the property $\ast :$ For all pairs of subsets $\{a_{1},\ldots ,a_{m}\},\{b_{1},\ldots ,b_{n}\}\subseteq K\backslash \{0\}, (\cap _{i=1}^{m}(a_{i})(\cap _{j=1}^{n}(b_{j}))=\cap _{i,j=1}^{m,n}(a_{i}b_{j})$ . We discuss what the appropriate names for G-Dedekind domains and related notions should be. We also make some observations about how the DCDs behave under localizations and polynomial ring extensions.

Principal Primitive Ideals in Quadratic Orders and Pell’s Equations

In this paper, we introduce a method of determining whether the primitive ideal is principal in a real quadratic order, depending on the solvability of Pell’s equation.

Radicals of principal ideals and the class group of a Dedekind domain

For a Dedekind domain $D$, let $\mathcal{P}(D)$ be the set of ideals of $D$ that are radical of a principal ideal. We show that, if $D,D'$ are Dedekind domains and there is an order isomorphism between $\mathcal{P}(D)$ and $\mathcal{P}(D')$, then the rank of the class groups of $D$ and $D'$ is the same.

Semigroups whose right ideals are finitely generated

AbstractWe call a semigroup$S$weakly right noetherianif every right ideal of$S$is finitely generated; equivalently,$S$satisfies the ascending chain condition on right ideals. We provide an equivalent formulation of the property of being weakly right noetherian in terms of principal right ideals, and we also characterize weakly right noetherian monoids in terms of their acts. We investigate the behaviour of the property of being weakly right noetherian under quotients, subsemigroups and various semigroup-theoretic constructions. In particular, we find necessary and sufficient conditions for the direct product of two semigroups to be weakly right noetherian. We characterize weakly right noetherian regular semigroups in terms of their idempotents. We also find necessary and sufficient conditions for a strong semilattice of completely simple semigroups to be weakly right noetherian. Finally, we prove that a commutative semigroup$S$with finitely many archimedean components is weakly (right) noetherian if and only if$S/\mathcal {H}$is finitely generated.

Canonical form of Involutory Matrices over the Domain of Principal Ideals with Respect to Similarity Transformations

Canonical form of Involutory Matrices over the Domain of Principal Ideals with Respect to Similarity Transformations

Lubin–Tate Deformation Spaces and Fields of Norms

We construct a tower of fields from the rings R n which parametrize pairs (X,λ), where X is a deformation of a fixed one-dimensional formal group 𝕏 of finite height h, together with a Drinfeld level-n structure λ. We choose principal prime ideals 𝔭 n |(p) in each ring R n in a compatible way and consider the field K n ′ obtained by localizing R n at 𝔭 n and passing to the field of fractions of the completion. By taking the compositum K n =K n ′ K 0 of K n ′ with the completion K 0 of a certain unramified extension of K 0 ′ , we obtain a tower of fields (K n ) n which we prove to be strictly deeply ramified in the sense of Scholl. When h=2 we also investigate the question of whether this is a Kummer tower.

Powers of Principal Q-Borel ideals

AbstractFix a poset Q on $\{x_1,\ldots ,x_n\}$ . A Q-Borel monomial ideal $I \subseteq \mathbb {K}[x_1,\ldots ,x_n]$ is a monomial ideal whose monomials are closed under the Borel-like moves induced by Q. A monomial ideal I is a principal Q-Borel ideal, denoted $I=Q(m)$ , if there is a monomial m such that all the minimal generators of I can be obtained via Q-Borel moves from m. In this paper we study powers of principal Q-Borel ideals. Among our results, we show that all powers of $Q(m)$ agree with their symbolic powers, and that the ideal $Q(m)$ satisfies the persistence property for associated primes. We also compute the analytic spread of $Q(m)$ in terms of the poset Q.

Pairs of domains where all intermediate domains satisfy S-ACCP

The rings considered in this paper are commutative with identity. If [Formula: see text] is a subring of a ring [Formula: see text], then we assume that [Formula: see text] contains the identity element of [Formula: see text]. Let [Formula: see text] be a multiplicatively closed subset (m.c. subset) of a ring [Formula: see text]. An increasing sequence of ideals [Formula: see text] of [Formula: see text] is said to be [Formula: see text]-stationary if there exist [Formula: see text] and [Formula: see text] such that [Formula: see text] for all [Formula: see text]. This paper is motivated by the research work [A. Hamed and H. Kim, On integral domains in which every ascending chain on principal ideals is [Formula: see text]-stationary, Bull. Korean Math. Soc. 57(5) (2020) 1215–1229]. Let [Formula: see text] be a m.c. subset of an integral domain [Formula: see text]. We say that [Formula: see text] satisfies [Formula: see text]-ACCP if every increasing sequence of principal ideals of [Formula: see text] is [Formula: see text]-stationary. Let [Formula: see text] be a subring of an integral domain [Formula: see text] and let [Formula: see text] be a m.c. subset of [Formula: see text]. We say that [Formula: see text] is an [Formula: see text]-ACCP pair if [Formula: see text] satisfies [Formula: see text]-ACCP for every subring [Formula: see text] of [Formula: see text] with [Formula: see text]. The aim of this paper is to provide some pairs of domains [Formula: see text] such that [Formula: see text] is an [Formula: see text]-ACCP pair, where [Formula: see text] is a m.c. subset of [Formula: see text].

On PF-spaces

Whenever, of any two zero-sets of a space X whose union is all of X at least one is open, then we call the space X a PF-space. We show that PF-spaces are precisely F-spaces in which all but at most one point is a P-point. As algebraic characterizations of PF-spaces, it is shown in this paper that X is a PF-space if and only if, of any two ideals in C(X) whose sum (intersection) is a proper z-ideal at least one of them is a z-ideal or equivalently, of any two ideals in C(X) whose product is a P-ideal, at least one is a P-ideal. These are also equivalent to saying that of any two comaximal principal ideals of C(X), one is semiprime and the other is convex. The relations between PF-spaces and other familiar spaces are investigated and we observe that a space is a cozero complemented PF-space if and only if it is a basically disconnected essential almost P-space (a space in which at most one point fails to be an almost P-point).

More on Prime, Maximal and Principal Soft Ideals of Soft Rings

In this paper, we aim to extend the studies [M. R. Alimoradi, R. Rezaei and M. Rahimi, Some notes on ideals in soft rings, Journal Australian Journal of Basic and Applied Sciences 6(3) (2012) 717–721; F. Koyuncu and B. Tanay, Some soft algebraic structures, Journal of New Results Science 10(2016) 38–51] as regards maximal, prime and principal soft ideal of soft rings, characterize soft rings with these soft ideals and also provide some more relations between maximal, prime and principal soft ideals of soft rings. The notions of maximality and primeness points of soft ideals of a soft rings are defined, maximal and prime idealistic soft rings as well as maximal, prime and principal soft ideal of soft rings and their basic properties are more investigated under certain conditions, especially by means of homomorphism and epimorphism of rings. We apply some of the basic results about maximal ideals and prime ideals in classical abstract algebra to maximal, prime and principal idealistic soft rings and we give some of their interrelations between each others.

On the integral domains characterized by a Bezout property on intersections of principal ideals

In this article we study two classes of integral domains. The first is characterized by having a finite intersection of principal ideals being finitely generated only when it is principal. The second class consists of the integral domains in which a finite intersection of principal ideals is always non-finitely generated except in the case of containment of one of the principal ideals in all the others. We relate these classes to many well-studied classes of integral domains, to star operations and to classical and new ring constructions.

On the key-exchange protocol using real quadratic fields

To prevent an exhaustive key-search attack of the key-exchange protocol using real quadratic fields, we need to ensure that the number l of reduced principal ideals in K is sufficiently large. In this paper we present an example of a family which are not valid for this protocol.

Structure of principal one-sided ideals

We give a thorough structural analysis of the principal one-sided ideals of arbitrary semigroups, and then apply this to full transformation semigroups and symmetric inverse monoids. One-sided ideals of these semigroups naturally occur as semigroups of transformations with restricted range or kernel.

Optimal constants in non-commutative Hölder inequality for quasi-norms

We resolve an open problem due to B. Simon [Trace ideals and their applications, Cambridge University Press, Cambridge–New York, 1979; Trace ideals and their applications, American Mathematical Society, Providence, RI, 2005] concerning optimal constants in Hölder inequality for certain quasi-normed principal ideals of importance in Mathematical Physics and Noncommutative Geometry.

Sharing a Measure of Maximal Entropy in Polynomial Semigroups

Abstract Let $P_1,P_2,\dots , P_k$ be complex polynomials of degree at least two that are not simultaneously conjugate to monomials or to Chebyshev polynomials, and $S$ the semigroup under composition generated by $P_1,P_2,\dots , P_k$. We show that all elements of $S$ share a measure of maximal entropy if and only if the intersection of principal left ideals $SP_1\cap SP_2\cap \dots \cap SP_k$ is non-empty.

Truncations of ordered abelian groups

We give axioms for a class of ordered structures, called truncated ordered abelian groups (TOAG’s) carrying an addition. TOAG’s come naturally from ordered abelian groups with a 0 and a +, but the addition of a TOAG is not necessarily even a cancellative semigroup. The main examples are initial segments [0, tau ] of an ordered abelian group, with a truncation of the addition. We prove that any model of these axioms (i.e. a truncated ordered abelian group) is an initial segment of an ordered abelian group. We define Presburger TOAG’s, and give a criterion for a TOAG to be a Presburger TOAG, and for two Presburger TOAG’s to be elementarily equivalent, proving analogues of classical results on Presburger arithmetic. Their main interest for us comes from the model theory of certain local rings which are quotients of valuation rings valued in a truncation [0, a] of the ordered group {mathbb {Z}} or more general ordered abelian groups, via a study of these truncations without reference to the ambient ordered abelian group. The results are used essentially in a forthcoming paper (D’Aquino and Macintyre, The model theory of residue rings of models of Peano Arithmetic: The prime power case, 2021, arXiv:2102.00295) in the solution of a problem of Zilber about the logical complexity of quotient rings, by principal ideals, of nonstandard models of Peano arithmetic.

Koszul multi-Rees algebras of principal L-Borel ideals

Given a monomial m in a polynomial ring and a subset L of the variables of the polynomial ring, the principal L-Borel ideal generated by m is the ideal generated by all monomials which can be obtained from m by successively replacing variables of m by those which are in L and have smaller index. Given a collection I={I1,…,Ir} where Ii is Li-Borel for i=1,…,r (where the subsets L1,…,Lr may be different for each ideal), we prove in essence that if the bipartite incidence graph among the subsets L1,…,Lr is chordal bipartite, then the defining equations of the multi-Rees algebra of I has a Gröbner basis of quadrics with squarefree lead terms under a lexicographic order. Thus the multi-Rees algebra of such a collection of ideals is Koszul, Cohen-Macaulay, and normal. This significantly generalizes a theorem of Ohsugi and Hibi on Koszul bipartite graphs. As a corollary we obtain that the multi-Rees algebra of a collection of principal Borel ideals is Koszul. To prove our main result we use a fiber-wise Gröbner basis criterion for the kernel of a toric map and we introduce a modification of Sturmfels' sorting algorithm.