Valuation and Pseudovaluation Subrings of an Integral Domain

Let R, S be two rings. We say that R is a valuation subring of S (R is a VD in S, for short) if R is a proper subring of S and whenever x ∈ S, we have x ∈ R or x ^?1 ∈ R. We denote by Nu(R) the set of all nonunit elements of a ring R. We say that R is a pseudovaluation subring of S (R is a PV in S, for short) if R is a proper subring of S and x ^?1 a ∈ R, for each x ∈ S?R, a ∈ Nu(R). This article deals with the study of valuation subrings and pseudovaluation subrings of a ring; interactions between the two notions are also given. Let R be a PV in S; the Krull dimension of the polynomial ring on n indetrminates over R is also computed.     

On the FIP Property for Extensions of Commutative Rings

ABSTRACT

A (unital) extension R ? T of (commutative) rings is said to have FIP (respectively be a minimal extension) if there are only finitely many (respectively no) rings S such that R ? S ? T. Transfer results for the FIP property for extensions of Nagata rings are obtained, including the following fact: if R ? T is a (module-) finite minimal ring extension, then R(X)?T(X) also is a (module-) finite minimal ring extension. The assertion obtained by replacing “is a (module-) finite minimal ring extension” with “has FIP” is valid if R is an infinite field but invalid if R is a finite field. A generalization of the Primitive Element Theorem is obtained by characterizing, for any field (more generally, any artinian reduced ring) R, the ring extensions R ? T which have FIP; and, if R is any field K, by describing all possible structures of the (necessarily minimal) ring extensions appearing in any maximal chain of intermediate rings between K and any such T. Transfer of the FIP and “minimal extension” properties is given for certain pullbacks, with applications to constructions such as CPI-extensions. Various sufficient conditions are given for a ring extension of the form R ? R[u], with u a nilpotent element, to have or not have FIP. One such result states that if R is a residually finite integral domain that is not a field and u is a nilpotent element belonging to some ring extension of R, then R ? R[u] has FIP if and only if (0 : u) ≠ 0. The rings R having only finitely many unital subrings are studied, with complete characterizations being obtained in the following cases: char(R)>0; R an integral domain of characteristic 0; and R a (module-)finite extension of ? which is not an integral domain. In particular, a ring of the last-mentioned type has only finitely many unital subrings if and only if (?:R)≠0. Some results are also given for the residually FIP property.     

The Relative Brauer Group of an Affine Double Plane

We study algebra classes and divisor classes on a normal affine surface of the form z ² = f(x, y). The affine coordinate ring is T = k[x, y, z]/(z ² ? f), and if R = k[x, y][f ^?1] and S = R[z]/(z ² ? f), then S is a quadratic Galois extension of R. If the Galois group is G, we show that the natural map H¹(G, Cl(T)) → H¹(G, Pic(S)) factors through the relative Brauer group B(S/R) and that all of the maps are onto. Sufficient conditions are given for H¹(G, Cl(T)) to be isomorphic to B(S/R). The groups and maps are computed for several examples.     

Decompositions of Quotient Rings and m-Power Commuting Maps

Let R be a semiprime ring with symmetric Martindale quotient ring Q, n ≥ 2 and let f(X) = X ⁿ h(X), where h(X) is a polynomial over the ring of integers with h(0) = ±1. Then there is a ring decomposition Q = Q ₁ ⊕ Q ₂ ⊕ Q ₃ such that Q ₁ is a ring satisfying S _2n?2, the standard identity of degree 2n ? 2, Q ₂ ? M _n (E) for some commutative regular self-injective ring E such that, for some fixed q > 1, x ^q  = x for all x ∈ E, and Q ₃ is a both faithful S _2n?2-free and faithful f-free ring. Applying the theorem, we characterize m-power commuting maps, which are defined by linear generalized differential polynomials, on a semiprime ring.     

The Largest Left Quotient Ring of a Ring

The left quotient ring (i.e., the left classical ring of fractions) Q_cl(R) of a ring R does not always exist and still, in general, there is no good understanding of the reason why this happens. In this article, existence of the largest left quotient ring Q_l(R) of an arbitrary ring R is proved, i.e., Q_l(R) = S₀(R)^?1R where S₀(R) is the largest left regular denominator set of R. It is proved that Q_l(Q_l(R)) = Q_l(R); the ring Q_l(R) is semisimple iff Q_cl(R) exists and is semisimple; moreover, if the ring Q_l(R) is left Artinian, then Q_cl(R) exists and Q_l(R) = Q_cl(R). The group of units Q_l(R)* of Q_l(R) is equal to the set {s^?1t | s, t ∈ S₀(R)} and S₀(R) = R ∩ Q_l(R)*. If there exists a finitely generated flat left R-module which is not projective, then Q_l(R) is not a semisimple ring. We extend slightly Ore's method of localization to localizable left Ore sets, give a criterion of when a left Ore set is localizable, and prove that all left and right Ore sets of an arbitrary ring are localizable (not just denominator sets as in Ore's method of localization). Applications are given for certain classes of rings (semiprime Goldie rings, Noetherian commutative rings, the algebra of polynomial integro-differential operators).     

On a Field-Theoretic Invariant for Extensions of Commutative Rings

Abstract

If R ? T is an extension of (commutative integral) domains, Λ(T/R) is defined as the supremum of the lengths of chains of intermediate fields in the extension k _R (Q ∩ R) ? k _T (Q), where Q runs over the prime ideals of T. The invariant Λ(T/R) is determined in case R and T are adjacent rings and in case Spec(R) = Spec(T) as sets. It is proved that if R is a domain with integral closure R′, then Λ(T/R) = 0 for all overrings T of R if and only if R′ is a Prüfer domain such that Λ(R′/R) = 0. If R ? T are domains such that the canonical map Spec(T) → Spec(R) is a homeomorphism (in the Zariski topology), then Λ(T/R) is bounded above by the supremum of the lengths of chains of rings intermediate between R and T. Examples are given to illustrate the sharpness of the results.     

Modules having Baer summands

Let R be an arbitrary ring with identity and M a right R-module with S = End_R(M). Let F be a fully invariant submodule of M and I^?1(F) denotes the set {m∈M:Im?F} for any subset I of S. The module M is called F-Baer if I^?1(F) is a direct summand of M for every left ideal I of S. This work is devoted to the investigation of properties of F-Baer modules. We use F-Baer modules to decompose a module into two parts consists of a Baer module and a module determined by fully invariant submodule F, namely, for a module M, we show that M is F-Baer if and only if M = F⊕N where N is a Baer module. By using F-Baer modules, we obtain some new results for Baer rings.     

Some Examples in the Theory of Factorization in Integral Domains

Let R be a UFD, S a subring of R and P be a nonzero commom ideal of R and S such that (i) R/P is a torsion-free S/P-module, (ii) if x,y ? S\P and c ? R with y = cx then c ? S. In this note we study the factorization properties of such subrings and thereby obtain some examples.     

A result concerning nilpotent values with generalized skew derivations on Lie ideals

ABSTRACT

Let n≥1 be a fixed integer, R a prime ring with its right Martindale quotient ring Q, C the extended centroid, and L a non-central Lie ideal of R. If F is a generalized skew derivation of R such that (F(x)F(y)?yx)ⁿ = 0 for all x,y∈L, then char(R) = 2 and R?M₂(C), the ring of 2×2 matrices over C.     

Right Centralizers of Semiprime Rings

Let R be a semiprime ring with Q _ml (R) the maximal left ring of quotients of R. Suppose that T: R → Q _ml (R) is an additive map satisfying T(x ²) = xT(x) for all x ∈ R. Then T is a right centralizer; that is, there exists a ∈ Q _ml (R) such that T(x) = xa for all x ∈ R.     

Subsets with Generalized Derivations Having Nilpotent Values on Lie Ideals

Let R be a prime ring, with no nonzero nil right ideal, Q the two-sided Martindale quotient ring of R, F a generalized derivation of R, L a noncommutative Lie ideal of R, and b ∈ Q. If, for any u, w ∈ L, there exists n = n(u, w) ≥1 such that (F(uw) ? bwu)ⁿ = 0, then one of the following statements holds:

1. F = 0 and b = 0;

2. R ? M₂(K), the ring of 2 × 2 matrices over a field K, b² = 0, and F(x) = ?bx, for all x ∈ R.

     

On Finite Maximal Chains of Weak Baer Going-Down Rings

Some recent results of Ayache on going-down domains and extensions of domains that either are residually algebraic or have DCC on intermediate rings are generalized to the context of extensions of commutative rings. Given a finite maximal chain 𝒞 of R-subalgebras of a weak Baer ring T, it is shown how a “min morphism” hypothesis can be used to transfer the “going-down ring” property from R to each member of 𝒞. The integral minimal ring extensions which are min morphisms are classified. The ring extensions satisfying FCP (i.e., for which each chain of intermediate rings is finite) are characterized as the strongly affine extensions with DCC on intermediate rings. In the relatively integrally closed case, such extensions R ? T induce open immersions Spec(S) → Spec(R) for each R-subalgebra S of T.     

Varieties of strange plane curves

Abstract

Let A be a commutative ring with identity, let X, Y be indeterminates and let F(X,Y), G(X, Y) ∈ A[X, Y] be homogeneous. Then the pair F(X, Y), G(X, Y) is said to be radical preserving with respect to A if Rad((F(x, y), G(x, y))R) = Rad((x,y)R) for each A-algebra R and each pair of elements x, y in R. It is shown that infinite sequences of pairwise radical preserving polynomials can be obtained by homogenizing cyclotomic polynomials, and that under suitable conditions on a ?-graded ring A these can be used to produce an infinite set of homogeneous prime ideals between two given homogeneous prime ideals P ? Q of A such that ht(Q/P) = 2.     

On the finiteness of a field-theoretic invariant for commutative rings

If T is a (commutative unital) ring extension of a ring R, then Λ(T /R) is defined to be the supremum of the lengths of chains of intermediate fields between R _P /P R _P and T _Q /QT _Q , where Q varies over Spec(T) and P:= Q ∩ R. The invariant σ(R):= sup Λ(T/R), where T varies over all the overrings of R. It is proved that if Λ(S/R)< ∞ for all rings S between R and T, then (R, T) is an INC-pair; and that if (R, T) is an INC-pair such that T is a finite-type R-algebra, then Λ(T/R)< ∞. Consequently, if R is a domain with σ(R) < ∞, then the integral closure of R is a Prüfer domain; and if R is a Noetherian G-domain, then σ(R) < ∞, with examples showing that σ(R) can be any given non-negative integer. Other examples include that of a onedimensional Noetherian locally pseudo-valuation domain R with σ(R)=∞.     

Invertibility of Matrices Over Subrings

Given rings R ? S, consider the division closure 𝒟(R, S) and the rational closure ?(R, S) of R in S. If S is commutative, then 𝒟(R, S) = ?(R, S) = RT ^?1, where T = {t ∈ R | t ^?1 ∈ S}. We show that this is also true if we assume only that R is commutative.     

On finitely generated module whose first nonzero fitting ideal is maximal

Let R be a Noetherian ring and M be a finitely generated R-module. Let I(M) be the first nonzero Fitting ideal of M. The main result of this paper asserts that when I(M) = Q is a regular maximal ideal of R, then M?R∕Q⊕P, for some projective R-module P of constant rank if and only if T(M)?QM. As a consequence, it is shown that if M is an Artinian R-module and I(M) = Q is a regular maximal ideal of R, then M?R∕Q.     

Rings Over which the Krull Dimension and the Noetherian Dimension of All Modules Coincide

We denote by 𝒜(R) the class of all Artinian R-modules and by 𝒩(R) the class of all Noetherian R-modules. It is shown that 𝒜(R) ? 𝒩(R) (𝒩(R) ? 𝒜(R)) if and only if 𝒜(R/P) ? 𝒩(R/P) (𝒩(R/P) ? 𝒜(R/P)), for all centrally prime ideals P (i.e., ab ∈ P, a or b in the center of R, then a ∈ P or b ∈ P). Equivalently, if and only if 𝒜(R/P) ? 𝒩(R/P) (𝒩(R/P) ? 𝒜(R/P)) for all normal prime ideals P of R (i.e., ab ∈ P, a, b normalize R, then a ∈ P or b ∈ P). We observe that finitely embedded modules and Artinian modules coincide over Noetherian duo rings. Consequently, 𝒜(R) ? 𝒩(R) implies that 𝒩(R) = 𝒜(R), where R is a duo ring. For a ring R, we prove that 𝒩(R) = 𝒜(R) if and only if the coincidence in the title occurs. Finally, if Q is the quotient field of a discrete valuation domain R, it is shown that Q is the only R-module which is both α-atomic and β-critical for some ordinals α,β ≥ 1 and in fact α = β = 1.     

Group homomorphisms inducing isomorphisms in cohomology

In this article, we call a ring R right generalized semiregular if for any a ∈ R there exist two left ideals P, L of R such that lr(a) = P⊕ L, where P ? Ra and Ra ∩ L is small in R. The class of generalized semiregular rings contains all semiregular rings and all AP-injective rings. Some properties of these rings are studied and some results about semiregular rings and AP-injective rings are extended. In addition, we call a ring R semi-π-regular if for any a ∈ R there exist a positive integer n and e ² = e ∈ a ⁿ R such that (1 ? e)a ⁿ  ∈ J(R), the Jacobson radical of R. It is shown that a ring R is semi-π-regular if and only if R/J(R) is π-regular and idempotents can be lifted modulo J(R).     

An Essential Extension with Nonisomorphic Ring Structures II

We construct a ring R with R = Q(R), the maximal right ring of quotients of R, and a right R-module essential extension S _R of R _R such that S has several distinct isomorphism classes of compatible ring structures. It is shown that under one class of these compatible ring structures, the ring S is not a QF-ring (in fact S is not even a right FI-extending ring), while under all other remaining classes of the ring structures, the ring S is QF. We demonstrate our results by an application to a finite ring.