Multiplicative closure operations on ring extensions

Star operations are defined by R. E. Hodel in 1994. In this paper some relations among star operators, sequential closure operators and closure operators are discussed. Moreover, we introduce an induced topology by a family of subsets of a space, and some interesting results about star operators are established by the induced topology.

Star operations on overrings of Noetherian domains

We introduce an operator on classes of regular languages, the star-free closure. Our motivation is to generalize standard results of automata theory within a unified framework. Given an arbitrary input class \(\mathscr{C}\) , the star-free closure operator outputs the least class closed under Boolean operations and language concatenation, and containing all languages of \(\mathscr{C}\) as well as all finite languages. We establish several equivalent characterizations of star-free closure: in terms of regular expressions, first-order logic, pure future and future-past temporal logic, and recognition by finite monoids. A key ingredient is that star-free closure coincides with another closure operator, defined in terms of regular operations where Kleene stars are allowed in restricted contexts. A consequence of this first result is that we can decide membership of a regular language in the star-free closure of a class whose separation problem is decidable. Moreover, we prove that separation itself is decidable for the star-free closure of any finite class, and of any class of group languages having itself decidable separation (plus mild additional properties). We actually show decidability of a stronger property, called covering.

Finite extensions of complex commutative Banach algebras are naturally related to corresponding finite covering maps between the carrier spaces for the algebras. In the case of function rings, the finite extensions are induced by the corresponding finite covering maps, and the topological properties of the coverings are strongly reflected in the algebraic properties of the extensions and conversely. Of particular interest to us is the class of finite covering maps for which the induced extensions of function rings admit primitive generators. This is exactly the class of polynomial covering maps and the extensions are algebraic extensions defined by the underlying Weierstrass polynomials.The purpose of this paper is to develop a suitable Galois theory for finite extensions of function rings induced by finite covering maps and to apply it in the case of Weierstrass polynomials and polynomial covering maps.

Let $A$ be a Noetherian ring and $I$ be a nonzero ideal of $A$. Let $\mathcal{R}(A, I)$ be the generalized Rees ring of the ideal $I$. Let $\sigma$ (resp. $\hat{\sigma}$ ) be a semiprime operation on the set of ideals of $A$ (resp. $\mathcal{R}(A, I)$ ). The enough integers $n$ under certain conditions. We first show in this paper, examples of semi-prime operations $\sigma$ and $\hat{\sigma}$ such that $\hat{\sigma}\left[u^n \mathcal{R}(A, I)\right] \cap A=\sigma\left(I^n\right)$ for all integers $n$, and reveal that $A_\sigma(I)=\left\{Q \cap A ; Q \in \operatorname{Ass}_{\mathcal{R}(A, I)}\left(\mathcal{R}(A, I) / \hat{\sigma}\left(u^n \mathcal{R}(A, I)\right)\right)\right\}$ when $n$ is large enough in $\mathbb{Z}$. Finally, we extend these results to filtrations. Received: December 12, 2023Revised: February 27, 2024Accepted: March 15, 2024

The Eakin–Nagata theorem examines the condition that the Noetherian property passes through each other between subrings and extension rings in 1968. Later, a noncommutative version of Eakin–Nagata theorem was also proved. Lam called this version Eakin–Nagata–Eisenbud theorem. In addition, Anderson and Dumitrescu introduced the $S$-Noetherian concept which is an extended notion of the Noetherian property on commutative rings in 2002. In this paper, we consider the $S$-variant of Eakin–Nagata–Eisenbud theorem for general rings by using $S$-Noetherian modules. We also show that every right $S$-Noetherian domain is right Ore, which is embedded into a division ring. For a right $S$-Noetherian ring, we obtain sufficient conditions for its right ring of fractions to be right $S$-Noetherian or right Noetherian. As applications, the $S$-variant of Eakin–Nagata–Eisenbud theorem is applied to the composite polynomial, composite power series and composite skew polynomial rings.

Let be a ring extension, and let be a strict star operation on the extension The extension is said to satisfy the ⋆-hash property if for any two distinct subsets and of where is the set of all S-regular ⋆-maximal ideals of R. We study ring extensions satisfying the star-hash property. Special attention is given to the case of Prüfer extensions.

Here we discuss the role of the conductor of a ring extension vis-a-vis the descent of projectivity and injectivity. Regarding the former, the first result says that an injective homomorphism of commutative rings descends projectivity if it does so modulo the conductor. The cheap version—that with noetherian hypotheses—of the descent of projectivity by a finite homomorphism due to Gruson then follows easily. A carbon copy—with the natural modification—of the descent of injectivity is also proved. The statement of the results follows very closely the lines of similar work of Ferrand ([2]) on flat modules rather than those of the remarkable [3]. 1* Conductor and projectivity* Throughout rings will be commutative with identity element. The price to lift the restriction of commutativity would be to load the exposition with expressions like two-sided ideal, bi-module, etc., without any real gain, in view of the fact that for the applications the commutativity is critical. The main result of this section is, almost word for word, the projective analogue of [2]. Rather than using the results already obtained there, at no cost, we will provide complete proofs based on simple calculations. THEOREM 1.1. Let h: A —> B be an injective homomorphism of rings, I an ideal of A and E an A-module. Then E is A-projective if and only if B ®^ E is B-projective and E/IE is A/I-projectiv e in the following cases: (i) I is also a B-ideaL (ii) I is nilpotent.

The purpose of this paper is to introduce two new classes of rings that is closely related to the classes of piecewise Noetherian domains and piecewise w-Noetherian domains. Let is a commutative ring and is a divided prime ideal of Let be a ring with total quotient ring T(R), and set such that for every and every Then is a ring homomorphism from T(R) into and restricted to R is also a ring homomorphism from R into given by for every At this setting, we introduce the notions of -piecewise Noetherian rings and -piecewise w-Noetherian rings and we show that the theories of -piecewise Noetherian (resp. -piecewise w-Noetherian) rings resemble those of piecewise Noetherian (resp. piecewise w-Noetherian) domains.

Uniquely separable extensions

The rings of the title are characterized. In view of earlier work on this problem, the main contribution here is the following result. Let R be a (commutative unital) ring extension of ℤ of the form ℤ[t] which is not an integral domain and which is not integral over ℤ. Then R has only finitely many (unital) subrings if and only if there exist nonzero integers a, b with at = b such that the minimal positive such a and the corresponding b satisfy (i) is integral over ℤ and (ii) there does not exist a prime number p such that ker(ϕ)⊆ p ℤ[X], where ϕ is the (unital) ring homomorphism ℤ[X] → R sending X to t. It is also proved that if T is any finite ring and n any nonzero integer, then T × ℤ[1/n] has only finitely many subrings.

Let R be a commutative unital ring and E a unital R-module. Then the canonical injective ring homomorphism from R into the idealization R(+) E is a minimal ring homomorphism if and only if E is a simple R-module. For E nonzero, R(+)E is not (R-algebra isomorphic to) an overring of R. If E 1 and E 2 are nonisomorphic simple R-modules, then R(+) E 1 and R(+) E 2 give minimal ring extensions of R which are not isomorphic as R-algebras. The ring of dual numbers over R is a minimal ring extension of R ⇔ R × R is a minimal ring extension of R ⇔ R is a field.

An integral domain R with quotient field Q is defined to be a Q-ring if Ext R 1 ⁡ ( Q , R ) ≅ Q \operatorname {Ext}_R^1(Q,R) \cong Q . It is shown that R is a Q-ring if and only if there exists an R-module A such that Hom R ( A , R ) = 0 {\operatorname {Hom}_R}(A,R) = 0 and Ext R 1 ⁡ ( A , R ) ≅ Q \operatorname {Ext}_R^1(A,R) \cong Q . If A is such an R-module and t ( A ) t(A) is its torsion submodule, then it is proved that A / t ( A ) A/t(A) necessarily has rank one. There are only three kinds of Q-rings, namely, Q 0 - , Q 1 - {Q_0}{\text {-}},{Q_1}{\text {-}} , or Q 2 {Q_2} -rings. These are described by the fact that if R is a Q-ring, then K = Q / R K = Q/R can only have 0, 1, or 2 proper h-divisible submodules. If H is the completion of R in the R-topology, then R is one of the three kinds of Q-rings if and only if H ⊗ R Q H{ \otimes _R}Q is one of the three possible kinds of 2-dimensional commutative Q-algebras. Examples of all three kinds of Q-rings are produced, and the behavior of Q-rings under ring extensions is examined. General conditions are given for a ring not to be a Q-ring. As an application of the theory, necessary and sufficient conditions are found for the integral closure of a non-complete Noetherian domain to be a complete discrete valuation ring.

A famous theorem of Kuratowski states that, in a topological space, at most 14 distinct sets can be produced by repeatedly applying the operations of closure and complement to a given set. We re-examine this theorem in the setting of formal languages, where closure is either Kleene closure or positive closure. We classify languages according to the structure of the algebra they generate under iterations of complement and closure. There are precisely 9 such algebras in the case of positive closure, and 12 in the case of Kleene closure. We study how the properties of being open and closed are preserved under concatenation. We investigate analogues, in formal languages, of the separation axioms in topological spaces; one of our main results is that there is a clopen partition separating two words if and only if the words do not commute. We can decide in quadratic time if the language specified by a DFA is closed, but if the language is specified by an NFA, the problem is PSPACE-complete.KeywordsTopological SpaceFormal LanguageClosure OperatorRegular LanguageShort WordThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Union-free expressions are used in union normal form to decompose any regular language to a finite union of union-free languages. Based on the automata characterisation of the union-free languages, by restricting the 1CFPAs not to have transitions by the empty word, or to be deterministic, the n-union-free and the deterministic union-free languages are defined. Union-complexity as a measure of descriptional complexity of regular languages was introduced recently. By the minimum number of union-free/n-union-free/deterministic union-free languages needed to get a regular language as their union, its union-complexity/n-union-complexity/d-union-complexity is defined. It is already known that union-complexity and n-union-complexity are finite for every regular language, however there are regular languages with infinite d-union-complexity. Operational union-complexity, that is, to predict the union-complexity of a language obtained by a language operation from languages with known union-complexity is an important and interesting question belonging to the field of descriptional complexity of formal systems. In the present paper, the Kleene plus, the positive Kleene closure operator is studied. As the Kleene star and plus operations have very different effects on the union-free languages, it is an interesting problem to investigate how the union-complexities may change under this operation. In particular, we show that the union-complexity of a regular language is not growing when this operation is being applied on it. On the other hand, the n-union-complexity of the Kleene plus of an n-union-free language remains 1, but the n-union-complexity of the Kleene plus of other regular languages may grow. Further, the deterministic union-complexity may jump to an infinite value even if the original language had a relatively small deterministic union-complexity, e.g., 4.Keywordsunion-complexityunion-free languagesregular expressionsKleene closure