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.     

Ideal-Symmetric and Semiprime Rings

Lambek extended the usual commutative ideal theory to ideals in noncommutative rings, calling an ideal A of a ring R symmetric if rst ∈ A implies rts ∈ A for r, s, t ∈ R. R is usually called symmetric if 0 is a symmetric ideal. This naturally gives rise to extending the study of symmetric ring property to the lattice of ideals. In the process, we introduce the concept of an ideal-symmetric ring. We first characterize the class of ideal-symmetric rings and show that this ideal-symmetric property is Morita invariant. We provide a method of constructing an ideal-symmetric ring (but not semiprime) from any given semiprime ring, noting that semiprime rings are ideal-symmetric. We investigate the structure of minimal ideal-symmetric rings completely, finding two kinds of basic forms of finite ideal-symmetric rings. It is also shown that the ideal-symmetric property can go up to right quotient rings in relation with regular elements. The polynomial ring R[x] over an ideal-symmetric ring R need not be ideal-symmetric, but it is shown that the factor ring R[x]/xⁿR[x] is ideal-symmetric over a semiprime ring R.     

Reflexive Property of Rings

Mason introduced the reflexive property for ideals, and then this concept was generalized by Kim and Baik, defining idempotent reflexive right ideals and rings. In this article, we characterize aspects of the reflexive and one-sided idempotent reflexive properties, showing that the concept of idempotent reflexive ring is not left-right symmetric. It is proved that a (right idempotent) reflexive ring which is not semiprime (resp., reflexive), can always be constructed from any semiprime (resp., reflexive) ring. It is also proved that the reflexive condition is Morita invariant and that the right quotient ring of a reflexive ring is reflexive. It is shown that both the polynomial ring and the power series ring over a reflexive ring are idempotent reflexive. We obtain additionally that the semiprimeness, reflexive property and one-sided idempotent reflexive property of a ring coincide for right principally quasi-Baer rings.     

Minimalities and Modules over Dedekind-Like Rings

We are interested in (right) modules M satisfying the following weak divisibility condition: If R is the underlying ring, then for every r ∈ R either Mr = 0 or Mr = M. Over a commutative ring, this is equivalent to say that M is connected with regular generics. Over arbitrary rings, modules which are “minimal” in several model theoretic senses satisfy this condition. In this article, we investigate modules with this weak divisibility property over Dedekind-like rings and over other related classes of rings.     

Free Pairs of Bicyclic Units in the Integral Group Ring of the Dihedral Group

Hirano studied the quasi-Armendariz property of rings, and then this concept was generalized by some authors, defining quasi-Armendariz property for skew polynomial rings and monoid rings. In this article, we consider unified approach to the quasi-Armendariz property of skew power series rings and skew polynomial rings by considering the quasi-Armendariz condition in mixed extension ring [R; I][x; σ], introducing the concept of so-called (σ, I)-quasi Armendariz ring, where R is an associative ring equipped with an endomorphism σ and I is an σ-stable ideal of R. We study the ring-theoretical properties of (σ, I)-quasi Armendariz rings, and we obtain various necessary or sufficient conditions for a ring to be (σ, I)-quasi Armendariz. Constructing various examples, we classify how the (σ, I)-quasi Armendariz property behaves under various ring extensions. Furthermore, we show that a number of interesting properties of an (σ, I)-quasi Armendariz ring R such as reflexive and quasi-Baer property transfer to its mixed extension ring and vice versa. In this way, we extend the well-known results about quasi-Armendariz property in ordinary polynomial rings and skew polynomial rings for this class of mixed extensions. We pay also a particular attention to quasi-Gaussian rings.     

Generalized APP-Rings

We say a ring R is (centrally) generalized left annihilator of principal ideal is pure (APP) if the left annihilator ? _R (Ra) ⁿ is (centrally) right s-unital for every element a ∈ R and some positive integer n. The class of generalized left APP-rings includes generalized left (principally) quasi-Baer rings and left APP-rings (and hence left p.q.-Baer rings, right p.q.-Baer rings, and right PP-rings). The class of centrally generalized left APP-rings is closed under finite direct products, full matrix rings, and Morita invariance. The behavior of the (centrally) generalized left APP condition is investigated with respect to various constructions and extensions, and it is used to generalize many results on generalized PP-rings with IFP and semiprime left APP-rings. Moreover, we extend a theorem of Kist for commutative PP rings to centrally generalized left APP rings for which every prime ideal contains a unique minimal prime ideal without using topological arguments. Furthermore, we give a complete characterization of a considerably large family of centrally generalized left APP rings which have a sheaf representation.     

Baer and Quasi-Baer Properties of Skew Generalized Power Series Rings

Let R be a ring, (S, ≤) a strictly ordered monoid and ω: S → End(R) a monoid homomorphism. The skew generalized power series ring R[[S, ω]] is a common generalization of (skew) polynomial rings, (skew) power series rings, (skew) Laurent polynomial rings, (skew) group rings, and Mal'cev–Neumann Laurent series rings. In this article, we study relations between the (quasi-) Baer, principally quasi-Baer and principally projective properties of a ring R, and its skew generalized power series extension R[[S, ω]]. As particular cases of our general results, we obtain new theorems on (skew) group rings, Mal'cev–Neumann Laurent series rings, and the ring of generalized power series.     

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.     

Transfer Results for the FIP and FCP Properties of Ring Extensions

For an extension E: R ? S of (commutative) rings and the induced extension F: R(X) ? S(X) of Nagata rings, the transfer of the FCP and FIP properties between E and F is studied. Then F has FCP ? E has FCP. The extensions E for which F has FIP are characterized. While E has FIP whenever F has FIP, the converse fails for certain subintegral extensions; it does hold if E is integrally closed, seminormal, or subintegral with R quasi-local having infinite residue field. If F has FIP, conditions are given for the sets of intermediate rings of E and F to be order-isomorphic.     

A Class of Auslander-Regular Macaulay Rings

Abstract

For a (left and right) noetherian semilocal ring R we analyse a regularity concept (called weak regularity) based on the equation gld R = dim R. Examples are regular Cohen-Macaulay orders over a regular local ring, localized enveloping algebras of finite dimensional Lie algebras, and the regular rings classified in Rump (2001b). We prove that weakly regular rings are Auslander-regular and Macaulay.     

When is a Sum of Annihilator Ideals an Annihilator Ideal?

We call a ring R a right SA-ring if for any ideals I and J of R there is an ideal K of R such that r(I) + r(J) = r(K). This class of rings is exactly the class of rings for which the lattice of right annihilator ideals is a sublattice of the lattice of ideals. The class of right SA-rings includes all quasi-Baer (hence all Baer) rings and all right IN-rings (hence all right selfinjective rings). This class is closed under direct products, full and upper triangular matrix rings, certain polynomial rings, and two-sided rings of quotients. The right SA-ring property is a Morita invariant. For a semiprime ring R, it is shown that R is a right SA-ring if and only if R is a quasi-Baer ring if and only if r(I) + r(J) = r(I ∩ J) for all ideals I and J of R if and only if Spec(R) is extremally disconnected. Examples are provided to illustrate and delimit our results.     

Cohen–Macaulay Modules and Holonomic Modules Over Filtered Rings

We study Gorenstein dimension and grade of a module M over a filtered ring whose associated graded ring is a commutative Noetherian ring. An equality or an inequality between these invariants of a filtered module and its associated graded module is the most valuable property for an investigation of filtered rings. We prove an inequality G?dim M ≤ G?dim gr M and an equality grade M = grade gr M, whenever Gorenstein dimension of gr M is finite (Theorems 2.3 and 2.8). We would say that the use of G-dimension adds a new viewpoint for studying filtered rings and modules. We apply these results to a filtered ring with a Cohen–Macaulay or Gorenstein associated graded ring and study a Cohen–Macaulay, perfect, or holonomic module.     

On a Hereditary Radical Property Relating to the Reducedness

In this note, a hereditary radical property, called homomorphically reduced rings, is introduced, observed, and applied. The dual concept of this property is also studied with the help of Courter, proving that any ring R (possibly without identity) has an ideal S such that S/K is not homomorphically reduced for each proper ideal K of S; and if L is an ideal of R with L ? S, then L/H is homomorphically reduced for some ideal H of R with H ? L. The concept of the homomorphical reducedness is shown to be equivalent to the left (right) weak regularity and the (strong) regularity for one-sided duo rings. It is proved that homomorphically reduced rings have several useful properties similar to those of (weakly) regular rings. It is proved that the homomorphical reducedness can go up to classical quotient rings. It is shown that if R is a reduced right Ore ring with the ascending chain condition (ACC) for annihilator ideals, then the maximal right quotient ring of R is strongly regular (hence homomorphically reduced).     

The McCoy Condition on Skew Polynomial Rings

Based on a theorem of McCoy on commutative rings, Nielsen called a ring R right McCoy if, for any nonzero polynomials f(x), g(x) over R, f(x)g(x) = 0 implies f(x)r = 0 for some 0 ≠ r ? R. In this note, we consider a skew version of these rings, called σ-skew McCoy rings, with respect to a ring endomorphism σ. When σ is the identity endomorphism, this coincides with the notion of a right McCoy ring. Basic properties of σ-skew McCoy rings are observed, and some of the known results on right McCoy rings are obtained as corollaries.     

Invariants of modular groups

For a faithful linear representation of a finite group G over a field of characteristic p, we study the ring of invariants. We especially study the polynomial and Cohen–Macaulay properties of the invariant ring. We first show that certain quotient rings of the invariant ring are polynomial rings by which we prove that the Hilbert ideal conjecture is true for a class of groups. In particular, we prove that the conjecture is true for vector invariant rings of Abelian reflection p-groups. Then we study the relationships between the invariant ring of G and that of a subgroup of G. Finally, we study the invariant rings of affine groups and show that, over a finite field, if an affine group contains all translations then the invariant ring is isomorphic to the invariant ring of a linear group.     

Power Series Rings Satisfying a Zero Divisor Property

In this note we continue to study zero divisors in power series rings and polynomial rings over general noncommutative rings. We first construct Armendariz rings which are not power-serieswise Armendariz, and find various properties of (power-serieswise) Armendariz rings. We show that for a semiprime power-serieswise Armendariz (so reduced) ring R with a.c.c. on annihilator ideals, R[[x]] (the power series ring with an indeterminate x over R) has finitely many minimal prime ideals, say B ₁,…,B _m , such that B ₁… B _m  = 0 and B _i  = A _i [[x]] for some minimal prime ideal A _i of R for all i, where A ₁,…,A _m are all minimal prime ideals of R. We also prove that the power-serieswise Armendarizness is preserved by the polynomial ring extension as the Armendarizness, and construct various types of (power-serieswise) Armendariz rings.     

A Property Satisfying Reducedness over Centers

This article concerns a ring property called pseudo-reduced-over-center that is satisfied by free algebras over commutative reduced rings.The properties of radicals of pseudo-reduced-over-center rings are investigated,especially related to polynomial rings.It is proved that for pseudo-reduced-over-center rings of nonzero characteristic,the centers and the pseudo-reduced-over-center property are preserved through factor rings modulo nil ideals.For a locally finite ring R,it is proved that if R is pseudo-reduced-over-center,then R is commutative and R/J(R) is a commutative regular ring with J(R) nil,where J(R) is the Jacobson radical of R.     

COMMUTATIVE RINGS WITH FINITE QUOTIENT FIELDS

Abstract

We consider the class of all commutative reduced rings for which there exists a finite subset T ? A such that all projections on quotients by prime ideals of A are surjective when restricted to T. A complete structure theorem is given for this class of rings,and it is studied its relation with other finiteness conditions on the quotients of a ring over its prime ideals.     

Elementary matrix reduction over certain rings

We explore elementary matrix reduction over certain rings characterized by properties related to stable range. Let R be a commutative ring. We call R locally stable if aR+bR = R??x∈R such that R∕(a+bx)R has stable range 1. We study locally stable rings and prove that every locally stable Bézout ring is an elementary divisor ring. Many known results on domains are thereby generalized.     

Notes on Generalized Hopfian and Weakly Co-Hopfian Modules

In this article, the generalized Hopficity and weakly co-Hopficity of modules over truncated polynomial and triangular matrix rings are considered. Also, it is obtained partially when the direct product R × S is a generalized Hopfian ring, where R and S are two rings.