Polynomial rings and weak second-order logic

By limiting the range of the predicate variables in a second-order language, one may obtain restricted versions of second-order logic such as weak second-order logic or definable subset logic. In this note, we provide an infinitary strongly complete axiomatization for several systems of this kind having the range of the predicate variables as a parameter. The completeness argument uses simple techniques from the theory of Boolean algebras. This article is dedicated to our friend John N. Crossley on the occasion of his 86th birthday.

Among commutative rings, the polynomial rings in a finite number of indeterminates enjoy important special properties and are frequently used in applications. As they are also of paramount importance in Algebraic Geometry, polynomial rings have been intensively studied. On the other hand, rings of formal power series have been extensively used in “algebroid geometry” and have many properties which are parallel to those of polynomial rings. In the first section of this chapter we shall define formal power series rings and we shall show that the main properties of polynomial rings which have been derived in previous chapters (see, in particular, Vol. I, Ch. I, §§ 16–18) hold also for formal power series rings. In the later sections of this chapter we shall give deeper properties of polynomial rings and, whenever possible, the parallel properties of power series rings.

Let R be a commutative ring with identity and $${\mathbb {N}}_0$$ be the additive monoid of nonnegative integers. We say that a function $$t : {\mathbb {N}}_0 \times {\mathbb {N}}_0 \rightarrow R$$ is a twist function on R if t satisfies the following three properties for all $$n, m, q \in {\mathbb {N}}_0$$ : (i) $$t(0,q) = 1$$ , (ii) $$t(n,m) = t(m,n)$$ , and (iii) $$t(n,m) \cdot t(n + m, q) = t (n, m + q) \cdot t(m, q)$$ . Let $$R[\![X]\!]$$ (resp., R[X]) be the set of power series (resp., polynomials) with coefficients in R. For $$f = \sum _{n=0}^{\infty } a_nX^n$$ and $$g = \sum _{n=0}^{\infty } b_nX^n \in R[\![X]\!]$$ , let $$f+g = \sum _{n=0}^{\infty } (a_n+b_n)X^n$$ , $$f*_tg = \sum _{n=0}^{\infty }(\sum _{i+j = n}t(i,j)a_ib_j)X^n$$ . Then, $$R^t[\![X]\!]:= (R[\![X]\!], +, *_t)$$ and $$R^t[X] := (R[X], +, *_t)$$ are commutative rings with identity that contain R as a subring. In this paper, we study ring-theoretic properties of $$R^t[\![X]\!]$$ and $$R^t[X]$$ with focus on divisibility properties including UFDs and GCD-domains. We also show how these two rings are related to the usual power series and polynomial rings.

In this paper, we introduce the concept of S‐‐ideals in both commutative and noncommutative rings. For a commutative ring R and a multiplicatively closed subset S, we show that many properties of ‐ideals apply to S‐‐ideals and examine their characteristics in various ring constructions, such as homomorphic image rings, quotient rings, cartesian product rings, polynomial rings, power series rings, idealization rings, and amalgamation rings. In noncommutative rings, where S is an m‐system, we define right S‐‐ideals. We demonstrate the equivalence of S‐‐ideals and right S‐‐ideals in commutative rings with identity and provide examples to illustrate the connections between right S‐prime ideals and ‐ideals.

It is well known that a polynomial [Formula: see text] over a commutative ring [Formula: see text] with identity is a zero-divisor in [Formula: see text] if and only if [Formula: see text] has a non-zero annihilator in the base ring, where [Formula: see text] is the polynomial ring with indeterminate [Formula: see text] over [Formula: see text]. But this result fails in non-commutative rings and in the case of formal power series ring. In this paper, we consider the problem of determining some annihilator properties of the formal power series ring [Formula: see text] over an associative non-commutative ring [Formula: see text]. We investigate relations between power series-wise McCoy property and other standard ring-theoretic properties. In this context, we consider right zip rings, right strongly [Formula: see text] rings and rings with right Property [Formula: see text]. We give a generalization (in the case of non-commutative ring) of a classical results related to the annihilator of formal power series rings over the commutative Noetherian rings. We also give a partial answer, in the case of formal power series ring, to the question posed in [1 Question, p. 16].

An Introduction to First-Order Logic

Formalizing Dijkstra's predicate transformer wp in weak second-order logic

Characterizing when the Rings R and R[T] Are Integrally Closed via Linear Equations over R

Let be a commutative ring with identity. Two elements and b in are called to be associates if and , or equivalently, if . The generalization of associate relation in R has given the idea for definitions of presimplifiable and weakly presimplifiable rings. First of all, it will be given definitions of very strong associate relation, strong regular associate relation, very strongly associate ring, and strongly regular associate ring. The presimplifiable ring is a commutative ring with the condition that every nonzero element is a unit element. While the weakly presimplifiable ring is a commutative ring with the condition that every nonzero element is regular element. Furthermore, it is shown that the relationship between very strongly associate ring with presimplifiable ring and the linkage between strongly regular associate ring and weakly presimplifiable ring. It is obtained that is a presimplifiable ring if and only if is a very strongly associate ring. Meanwhile, is a weakly presimplifiable ring if and only if is a strongly regular associate ring. Then, it is shown that the correlation between presimplifiable and weakly presimplifiable rings to its polynomial ring and its the formal power series ring . If is a weakly presimplifiable ring, then and are also weakly presimplifiable rings. However, if is a presimplifiable ring, then is also a presimplifiable ring but always not valid for .

Several different generalizations of finite factorization domains (i.e., integral domains where every nonzero nonunit has only finitely many divisors up to associates) have been defined for commutative rings with zero divisors. We study these notions in the context of commutative monoid rings with zero divisors, utilizing semigroup theory to simultaneously generalize and extend many past results about “finite factorization” properties in commutative polynomial rings. Along the way, we expand upon the general theory of factorization in commutative rings with zero divisors, providing new characterizations and results about several kinds of “finite factorization rings.”

Given a certain factorization property of a ring [Formula: see text], we can ask if this property extends to the polynomial ring over [Formula: see text] or vice versa. For example, it is well known that [Formula: see text] is a unique factorization domain if and only if [Formula: see text] is a unique factorization domain. If [Formula: see text] is not a domain, this is no longer true. In this paper, we survey unique factorization in commutative rings with zero divisors, and characterize when a polynomial ring over an arbitrary commutative ring has unique factorization.

Let [Formula: see text] be a commutative ring with identity, [Formula: see text] be an indeterminate over [Formula: see text], [Formula: see text] be the polynomial ring over [Formula: see text], [Formula: see text] be the power series ring over [Formula: see text], and [Formula: see text] be a principal prime ideal of [Formula: see text] with [Formula: see text]. It is well known that if [Formula: see text] is an integral domain, then [Formula: see text] is a DVR and [Formula: see text] has infinitely many such principal prime ideals. In this paper, among other things, we show that (i) [Formula: see text] is a field, (ii) [Formula: see text] is a DVR, but (iii) there is a ring [Formula: see text] such that [Formula: see text] has no principal prime ideal. We also study the maximal ideals of [Formula: see text] that are principal.

The polynomial ring over a Goldie ring need not be a Goldie ring

Let R be a commutative ring with identity. An element is said to be absolutely irreducible in R if for all natural numbers n > 1, rn has essentially only one factorization namely If is irreducible in R but for some n > 1, rn has other factorizations distinct from then r is called non-absolutely irreducible. In this paper, we construct non-absolutely irreducible elements in the ring of integer-valued polynomials. We also give generalizations of these constructions.

In this paper, we consider a basic question in commutative algebra: if I and J are ideals of a commutative ring S, when does IJ=I∩J? More precisely, our setting will be in a polynomial ring k[x0, …, xn], and the ideals I and J will define subschemes of the projective space Pnk over k. In this situation, we are able to relate the equality of product and intersection to the behavior of the cohomology modules of the subschemes defined by I and J. By doing this, we are able to prove several general algebraic results about the defining ideals of certain subschemes of projective space. Our main technique in this paper is a study of the deficiency modules of a subscheme V of Pn. These modules are important algebraic invariants of V, and reflect many of the properties of V, both geometric and algebraic. For instance, when V is equidimensional and dim V⩾1, the deficiency modules of V are invariant (up to a shift in grading) along the even liaison class of V [14, 11, 15, 7], although they do not in general completely determine the even liaison class, except in the case of curves in P3 [14]. On the algebraic side, at least for curves in P3, the deficiency modules have been shown to have connections to the number and degrees of generators of the saturated ideal defining V [12]. One of our main goals in this paper is to extend these results to subschemes of arbitrary codimension in any projective space Pn. We now describe the contents of this paper more precisely. In the first section, we set up our notation and give the basic definitions which we will use. Then we prove our main technical result: if I and J define subschemes V and Y, respectively, of Pn, we relate the quotient module (I∩J)/IJ to the cohomology of V, at least when V and Y meet properly. We are then able to give a different proof of a general statement due to Serre about when there is an equality of intersection and product. In the second section, we give an extension of Dubreil's Theorem on the number of generators of ideals in a polynomial ring. Specifically, our generalization works for an ideal I defining a locally Cohen–Macaulay, equidimensional subscheme V of any codimension in Pn, and relates the number of generators of the defining ideal to the length of certain Koszul homologies of the cohomology of V. The results in this section depend crucially on the identification done in Section 1 of the intersection modulo the product. Finally, in Section 3, we give an extension of a surprising result of Amasaki [1] showing a lower bound for the least degree of a minimal generator of the ideal of a Buchsbaum subscheme. Originally, Amasaki gave a bound in the case of Buchsbaum curves in P3 (and later gave a natural extension to Buchsbaum codimension 2 subschemes of Pn [2]). Easier proofs were subsequently given by Geramita and Migliore in [6], based on a determination of the free resolution of the ideal from a resolution of its deficiency module. For Buchsbaum codimension 2 subschemes of Pn whose intermediate cohomology vanishes, we are able to extend these considerations.