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In this paper, we study local rings from the perspective of reverse mathematics. We define local rings in a first-order way by using Π20 properties of invertible elements, where for a ring R possibly not commutative, R is left (resp. right) local if for any non-left (resp. non-right) invertible elements x, y ∈ R, x + y is not left (resp. right) invertible; R is local if for any non-invertible elements x, y ∈ R, x + y is not invertible. Firstly, we solve a question of Sato on characterizations of commutative local rings in his PhD thesis (Question 6.22 in Sato (2016)) and prove that the statement "a commutative ring is local if and only if it has at most one maximal ideal" is equivalent to ACA0 over RCA0. We also obtain a nice corollary in computable mathematics, i.e., there is a computable non-local ring with exactly two maximal ideals such that each of them Turing computes the Halting set K. Secondly, we study the equivalence among left local rings, right local rings, and local rings, showing that these three kinds of first-order local rings are equivalent over the weak basis theory RCA0. Finally, we extend the results of reverse mathematics on commutative local rings to noncommutative rings. [ABSTRACT FROM AUTHOR]

This paper explores the ideals and their structural properties in two generalizations of the partial transformation semigroup. Furthermore, principal, maximal, and minimal ideals within these semigroups are elucidated. [ABSTRACT FROM AUTHOR]

The existence of a maximal ideal in a general nontrivial commutative ring is tied together with the axiom of choice. Following Berardi, Valentini and thus Krivine but using the relative interpretation of negation (that is, as "implies 0 = 1 ") we show, in constructive set theory with minimal logic, how for countable rings one can do without any kind of choice and without the usual decidability assumption that the ring is strongly discrete (membership in finitely generated ideals is decidable). By a functional recursive definition we obtain a maximal ideal in the sense that the quotient ring is a residue field (every noninvertible element is zero), and with strong discreteness even a geometric field (every element is either invertible or else zero). Krull's lemma for the related notion of prime ideal follows by passing to rings of fractions. By employing a construction variant of set-theoretic forcing due to Joyal and Tierney, we expand our treatment to arbitrary rings and establish a connection with dynamical algebra: We recover the dynamical approach to maximal ideals as a parametrized version of the celebrated double negation translation. This connection allows us to give formal a priori criteria elucidating the scope of the dynamical method. Along the way we do a case study for proofs in algebra with minimal logic, and generalize the construction to arbitrary inconsistency predicates. A partial Agda formalization is available at an accompanying repository.1 See https://github.com/iblech/constructive-maximal-ideals/. This text is a revised and extended version of the conference paper (In Revolutions and Revelations in Computability. 18th Conference on Computability in Europe (2022) Springer). The conference paper only briefly sketched the connection with dynamical algebra; did not compare this connection with other flavors of set-theoretic forcing; and sticked to the case of commutative algebra only, passing on the generalization to inconsistency predicates and well-orders. [ABSTRACT FROM AUTHOR]

In this paper, we study the structure of the skew polynomial ring R = ( p + u p) [ x ; ] and its quotient ring R n = R / 〈 x n − 1 〉 , where p is an odd prime number, u 2 = 0 , and (u) = − u. We give an explicit structure of the ideals in R and R n and propose an algorithm to characterize them. We identify the structure of prime, maximal, and primary ideals in these rings. In particular, we prove that this group ring is not Laskerian. [ABSTRACT FROM AUTHOR]

The ring of periodic distributions on R d with usual addition of distributions, and with convolution is considered. Via Fourier series expansions, this ring is isomorphic to the ring S ′ (Z d) of all maps f : Z d → C of at most polynomial growth (that is, there exist a real number M > 0 and an integer m ≥ 0 such that | f (n) | ≤ M (1 + | n 1 | + ⋯ + | n d |) m for all n = (n 1 , ⋯ , n d) ∈ Z d ), with pointwise operations. It is shown that finitely generated ideals in S ′ (Z d) are principal, and ideal membership is characterised analytically. Calling an ideal in S ′ (Z d) fixed if there is a common index n ∈ Z d where each member vanishes, the fixed maximal ideals are described, and it is shown that not all maximal ideals are fixed. It is shown that finitely generated (hence principal) prime ideals in S ′ (Z d) are fixed maximal ideals. The Krull dimension of S ′ (Z d) is proved to be infinite, while the weak Krull dimension is shown to be equal to 1. [ABSTRACT FROM AUTHOR]

AbstractLet R be a ring with unity. The upper ideal relation graph [Formula: see text] of the ring R is the simple undirected graph whose vertex set is the set of all non-unit elements of R and two distinct vertices x, y are adjacent if and only if there exists a non-unit element [Formula: see text] such that the ideals (x) and (y) contained in the ideal (z). In this article, we obtain the girth, minimum degree and the independence number of [Formula: see text]. We obtain a necessary and sufficient condition on R, in terms of the cardinality of their principal ideals, such that the graph [Formula: see text] is planar and outerplanar, respectively. For a non-local commutative ring [Formula: see text], where Ri is a local ring with maximal ideal [Formula: see text] and [Formula: see text], we prove that the graph [Formula: see text] is perfect if and only if [Formula: see text] and each [Formula: see text] is a principal ideal. We also discuss all the finite rings R such that the graph [Formula: see text] is Eulerian. Moreover, we obtain the metric dimension and strong metric dimension of [Formula: see text], when R is a reduced ring. Finally, we determine the vertex connectivity, automorphism group, Laplacian and the normalized Laplacian spectrum of [Formula: see text]. We classify all the values of n for which the graph [Formula: see text] is Hamiltonian.

The injective topic is well known for its influential relevance in module theory and scholars have worked hard to identify generalizations for it. One of these generalizations is M-Mininjective and mininjective, so we extend these notions to S-act theory as well, since act theory represents the generalization of module theory. If every S-homomorphism from a simple M-cyclic subact of M S into N S can be extended to M S , an S-act N S is called M-mininjective. An S-act M S is referred to as mininjective, if for each simple right ideal A of S and every S-homomorphism from A into M S can be extended to S-homomorphism from S into M S . We looked at the properties and characterizations of S-act where all subacts are M-cyclic and simple and all subacts are merely simple. These topics are shown using examples. With the provided concepts, we were able to accomplish improved results, obtaining novel characterizations of mininjective acts in terms of duality conditions. Additionally, the conditions under which subacts inherit the mininjective and M-mininjective properties are studied. The connection between the act of maximal right ideals of S and the act of minimal subact of T M is explicated. Finally, the conditions under which the classes of M-mininjective acts and the classes of mininjective S-acts will coincide are defined. Our work's conclusions have been explained. [ABSTRACT FROM AUTHOR]

In this paper, we study the maximal ideals in a commutative ring of bicomplex numbers and then we describe the maximal ideals in a bicomplex algebra. We found that the kernel of a nonzero multiplicative B C -linear functional in a commutative bicomplex Banach algebra need not be a maximal ideal. Finally, we introduce the notion of bicomplex division algebra and generalize the Gelfand–Mazur theorem for the bicomplex division Banach algebra. [ABSTRACT FROM AUTHOR]

Significant stochastic orders can be characterized by other stochastic orders when probabilities are transformed by means of measurable mappings which fulfils certain conditions. The main aim of this manuscript is to provide a unified approach of those orders. Our study shows how to obtain important results of such orders by means of conditions on the underlying stochastic orders. Those results are mainly in relation to maximal generators, to transition kernels between ordered probabilities, and to probabilistic operators of kernels. [ABSTRACT FROM AUTHOR]

In this paper, we will now develop the content of quasi-commutative near_subtraction semi groups, Semi prime ideals, Proper prime ideals, globally idempotent principal ideals. Relation between proper prime ideals and semi prime ideals. [ABSTRACT FROM AUTHOR]

In an Almost Distributive Fuzzy Lattice (ADFL), the concept of S-ideals is introduced, and some significant features of these ideals are derived. S-ideals are used to categorise ADFLs. Prime ideals were also confirmed in ADFL. For the class of all S-ideals to become a DFL to the lattice of all ideals, a set of analogous requirements is stated, leading to the characterization of an ADFL. [ABSTRACT FROM AUTHOR]

The aim of this paper is twofold. First, to provide a characterization of clopen topologies. We show that a topology is clopen if and only if it is symmetric and Alexandroff. Second, to investigate when some special unitary subrings of a power set define a (clopen) topology. [ABSTRACT FROM AUTHOR]

This paper presents a PVS development of relevant results of the theory of rings. The PVS theory includes complete proofs of the three classical isomorphism theorems for rings, and characterizations of principal, prime and maximal ideals. Algebraic concepts and properties are specified and formalized as generally as possible allowing in this manner their application to other algebraic structures. The development provides the required elements to formalize important algebraic theorems. In particular, the paper presents the formalization of the general algebraic-theoretical version of the Chinese remainder theorem (CRT) for the theory of rings, as given in abstract algebra textbooks, proved as a consequence of the first isomorphism theorem. Also, the PVS theory includes a formalization of the number-theoretical version of CRT for the structure of integers, which is the version of CRT found in formalizations. CRT for integers is obtained as a consequence of the general version of CRT for the theory of rings. [ABSTRACT FROM AUTHOR]

We give a computational interpretation to an abstract formulation of Krull's theorem, by analysing its classical proof based on Zorn's lemma. Our approach is inspired by proof theory, and uses a form of update recursion to replace the existence of maximal ideals. Our main result allows us to derive, in a uniform way, algorithms which compute witnesses for existential theorems in countable abstract algebra. We give a number of concrete examples of this phenomenon, including the prime ideal theorem and Krull's theorem on valuation rings. [ABSTRACT FROM AUTHOR]