We prove a new extension result for $QB-$rings that allows us to examine extensions of rings where the ideal is purely infinite and simple. We then use this result to explore various constructions that provide new examples of $QB-$rings. More concretely, we show that a surjective pullback of two $QB-$rings is usually again a $QB-$ring. Specializing to the case of an extension of a semi-prime ideal $I$ of a unital ring $R$, the pullback setting leads naturally to the study of rings whose multiplier rings are $QB-$rings. For a wide class of regular rings, we give necessary and sufficient conditions for their multiplier rings to be $QB-$rings. Our analysis is based on the study of extensions and the use of non-stable $K-$theoretical techniques.
We define pullback and separated presentations of modules over pullback rings, and, if the ring is a pullback of epimorphisms over a semisimple ring, an algorithm reducing such a presentation of a module to an $R$-diagram. The latter is the input for a classification algorithm of finitely generated modules over a pullback ring of two Dedekind domains. As an example we show how to obtain an $R$-diagram for homology of a chain complex of free modules over a $p$-pullback ring.
The classical theorem of Milnor on pullback rings states that the category of projective modules over a pullback ring is equivalent to a certain category of gluing triples consisting of projective modules. We prove an analogous result on the level of derived categories, where the equivalence has to be replaced by an epivalence.
We use pullbacks of rings to realize the submonoids $M$ of $(\N_0\cup\{\infty\})^k$ which are the set of solutions of a finite system of linear diophantine inequalities as the monoid of isomorphism classes of countably generated projective right $R$-modules over a suitable semilocal ring. For these rings, the behavior of countably generated projective left $R$-modules is determined by the monoid $D(M)$ defined by reversing the inequalities determining the monoid $M$. These two monoids are not isomorphic in general. As a consequence of our results we show that there are semilocal rings such tha…
The aim of this paper is to investigate a general notion of $σ$-PBW extensions over Armendariz rings. As an application, the properties of being Baer, quasi-Baer, p.p. and p.q.-Baer are established for these extensions. We generalize several results in the literature for Ore extensions of injective type and skew PBW extensions.
Let $N$ and $H$ be groups, and let $G$ be an extension of $H$ by $N$. In this article we describe the structure of the complex group ring of $G$ in terms of data associated with $N$ and $H$. In particular, we present conditions on the building blocks $N$ and $H$ guaranteeing that $G$ satisfies the zero-divisor and idempotent conjectures. Moreover, for central extensions involving amenable groups we present conditions on the building blocks guaranteeing that the Kadison-Kaplansky conjecture holds for the group C*-algebra of $G$.
This paper investigates coherent-like conditions and related properties that a trivial extension might inherit from the ground ring over some classes of modules. It captures previous results dealing primarily with coherence, and also establishes satisfactory analogues of well-known coherence-like results on pullback constructions. Our results generate new families of examples of rings (with zerodivisors) subject to a given coherent-like condition.
Given two rings $R \subseteq S$, $S$ is said to be a minimal ring extension of $R$ if $R$ is a maximal subring of $S$. In this article, we study minimal extensions of an arbitrary ring $R$, with particular focus on those possessing nonzero ideals that intersect $R$ trivially. We will also classify the minimal ring extensions of prime rings, generalizing results of Dobbs, Dobbs & Shapiro, and Ferrand & Olivier on commutative minimal extensions.
We show that there exist noncommutative Ore extensions in which every right ideal is two-sided. This answers a problem posed by Marks in Duo Rings and Ore extensions, J.Algebra 280(2), (2004). We also provide an easy construction of one sided duo rings.
We investigate and characterize several kinds of elements such as units, idempotents, von Neumann regular, $π$-regular and clean elements for skew PBW extensions over weak compatible rings. We also study the notions of Gelfand and Harmonic rings for these families of algebras. The results presented here extend those corresponding in the literature for commutative and noncommutative rings of polynomial type.
A ring $R$ is said to be right McCoy if the equation $f(x)g(x)=0,$ where $f(x)$ and $g(x)$ are nonzero polynomials of $R[x],$ implies that there exists nonzero $s \in R$ such that $f(x)s = 0$. It is proven that no proper (triangular) matrix ring is one-sided McCoy. If there exists the classical right quotient ring $Q$ of a ring $R$, then $R$ is right McCoy if and only if $Q$ is right McCoy. It is shown that for many polynomial extensions, a ring $R$ is right McCoy if and only if the polynomial extension over $R$ is right McCoy. Other basic extensions of right McCoy rings are also studied.\left…
It is shown that there is a close relationship between ideal extensions of rings and trusses, that is, sets with a semigroup operation distributing over a ternary abelian heap operation. Specifically, a truss can be associated to every element of an extension ring that projects down to an idempotent in the extending ring; every weak equivalence of extensions yields an isomorphism of corresponding trusses. Furthermore, equivalence classes of ideal extensions of rings by integers are in one-to-one correspondence with associated trusses up to isomorphism given by a translation. Conversely, to any…
In this paper, using the general Mal'cev-Neumann construction of Laurent series rings, we construct a Laurent series ring with a base ring which is an extension of the field $\mathbb{Q}$ of rational numbers. Further, we establish some useful properties of such a ring.
Let $R$ be a ring and $S=R[x;σ,δ]$ its Ore extension. We prove under some conditions that $R$ is a quasi-Baer ring if and only if the Ore extension $R[x;σ,δ]$ is a quasi-Baer ring. Examples are provided to illustrate and delimit our results.
Let $R$ be a ring, $σ$ an injective endomorphism of $R$ and $δ$ a $σ$-derivation of $R$. We prove that if $R$ is semiprime left Goldie then the same holds for the Ore extension $R[x;σ,δ]$ and both rings have the same left uniform dimension.
Every quasi completely regular semiring is a b-lattice of completely Archimedean semirings, i.e., a b-lattice of nil-extensions of completely simple semirings. In this paper we consider the semiring which is a b-lattice of nil-extensions of orthodox completely simple semirings.
In this paper we discuss under which conditions cyclic essential extensions of simple modules over a differential operator ring R[z;d] are Artinian. In particular, we study the case when R is either d-simple or d-primitive. Furthermore, we obtain important results when R is an affine algebra of Kull dimension 2. As an application we characterize the differential operator rings C[x,y][z;d] for which cyclic essential extensions of simple modules are Artinian.
In this paper we define and study quasipolar general rings (with or without identity) and extend many of the basic results to the wider class. We obtain some new characterizations of quasipolar and strongly $π$-regular elements by using quasipolar general rings. We see that quasipolar general rings lies between strongly $π$-regular and strongly clean general rings. Consequently, we prove that $R$ is pseudopolar if and only if $R$ is strongly $π$-rad clean and quasipolar.
Let $R$ be a ring and $(σ,δ)$ a quasi-derivation of $R$. In this paper, we show that if $R$ is an $(σ,δ)$-skew Armendariz ring and satisfies the condition $(\mathcal{C_σ})$, then $R$ is right p.q.-Baer if and only if the Ore extension $R[x;σ,δ]$ is right p.q.-Baer. As a consequence we obtain a generalization of \cite{hong/2000}.
Many rings and algebras arising in quantum mechanics can be interpreted as skew PBW (Poincaré-Birkhoff-Witt) extensions. Indeed, Weyl algebras, enveloping algebras of finite-dimensional Lie algebras (and its quantization), Artamonov quantum polynomials, diffusion algebras, Manin algebra of quantum matrices, among many others, are examples of skew PBW extensions. In this short paper we study the d-Hermite condition about stably free modules for skew PBW extensions. For this purpose, we estimate the stable rank of these non-commutative rings. In addition, and close related with these questions, …
In this paper we consider centralizers of single elements in Ore extensions of the ring of polynomials in one variable over a field. We show that they are commutative and finitely generated as an algebra. We also show that for certain classes of elements their centralizer is singly generated as an algebra.
Let $X$ be a set of noncommuting variables of cardinality $card(X)\geqslant 2$, and ${\mathscr G}=\{σ_x\}_{x\in X}$, ${\mathscr D}=\{δ_x\}_{x\in X}$ be families of automorphisms and skew derivations of the ring $R$. It is proved that if the ring $R$ is semiprime Goldie, then the free skew extension $R[X;{\mathscr G},{\mathscr D}]$ is semiprimitive.
A ring $R$ is said to be an almost Armendariz ring if whenever product of two polynomials in $R[x]$ is zero, then product of their coefficients are in $N_{*}(R)$. In this article, for an endomorphism $α$ on $R$, we define an $α$-almost Armendariz ring of $R$ considering the polynomials in skew polynomial ring $R[x; α]$ instead of $R[x]$. It is the generalisation of an almost Armendariz ring [9] and an $α$-Armendariz ring [4]. Moreover, for an endomorphism $α$ of $R$, we define an $α$-skew almost Armendariz ring, and prove that a reversible ring $R$ with certain condition on endomorphism $α$, i…
A Galois field extension $E/F$ whose Galois group is a pro-2-group of an exponent of at most 8, with a nilpotency class of at most 4, is determined, such that it contains essential information about the Witt rings of all quadratic extensions of $F$.
A fundamental result of Springer says that a quadratic form over a field of characteristic not 2 is isotropic if it is so after an odd degree extension. In this paper we generalize Springer's theorem as follows. Let R be a an arbitrary semilocal ring, let S be a finite R-algebra of constant odd degree, which is {é}tale or generated by one element, and let q be a nonsingular R-quadratic form whose base ring extension q S is isotropic. We show that then q is already isotropic.
We apply set-theoretic methods to study projective modules and their generalizations over transfinite extensions of simple artinian rings R. When R is hereditary and of cardinality at most $2^ω$, we prove that the Weak Diamond and CH imply that projectivity of an arbitrary module can be tested at the layer epimorphisms of R.
In this paper we introduce the notion of weak $Σ$-rigid ring which extends $α$-rigid rings and $Σ$-rigid rings defined for Ore extensions and skew PBW extensions, respectively. We also present the notion of weak $Σ$-skew Armendariz ring which extends that of $α$-skew Armendariz ring. In this way we generalize several results in the literature, from Ore extensions of injective type to the more general setting of skew PBW extensions.
In this paper, we study the reflexive-nilpotents-property (briefly, RNP) for Ore extensions of injective type, and more generally, skew PBW extensions. With this aim, we introduce the notions of $Σ$-skew CN rings and $Σ$-skew reflexive (RNP) rings, for $Σ$ a finite family of ring endomorphisms of a ring $R$. Under certain conditions of compatibility, we study the transfer of the $Σ$-skew RNP property from a ring of coefficients to an Ore extension or skew PBW extension over this ring. We also consider this property for localizations of these noncommutative rings. Our results extend those corre…
In this paper we discuss some open problems of non-commutative algebra and non-commutative algebraic geometry from the approach of skew $PBW$ extensions and semi-graded rings. More exactly, we will analyze the isomorphism arising in the investigation of the Gelfand-Kirillov conjecture about the commutation between the center and the total ring of fractions of an Ore domain. The Serre's conjecture will be discussed for a particular class of skew $PBW$ extensions. The questions about the noetherianity and the Zariski cancellation property of Artin-Schelter regular algebras will be reformulated f…
Two observations in support of the thesis that trusses are inherent in ring theory are made. First, it is shown that every equivalence class of a congruence relation on a ring or, equivalently, any element of the quotient of a ring $R$ by an ideal $I$ is a paragon in the truss $\mathrm{T}(R)$ associated to $R$. Second, an extension of a truss by a one-sided module is described. Even if the extended truss is associated to a ring, the resulting object is a truss, never a ring, unless the module is trivial. On the other hand, if the extended truss is associated to a brace, the resulting truss is …
Given a ring $R$, we study the bimodules $M$ for which the trivial extension $R\propto M$ is morphic. We obtain a complete characterization in the case where $R$ is left perfect, and we prove that $R\propto Q/R$ is morphic when $R$ is a commutative reduced ring with classical ring of quotients $Q$. We also extend some known results concerning the connection between morphic rings and unit regular rings.
In this paper, we introduce the concept of $Σ$-semicommutative ring, for $Σ$ a finite family of endomorphisms of a ring $R$. We relate this class of rings with other classes of rings such that Abelian, reduced, $Σ$-rigid, nil-reversible and rings satisfying the $Σ$-skew reflexive nilpotent property. Also, we study some ring-theoretical properties of skew PBW extensions over $Σ$-semicommutative rings. We prove that if a ring $R$ is $Σ$-semicommutative with certain conditions of compatibility on derivations, then for every skew PBW extension $A$ over $R$, $R$ is Baer if and only if $R$ is quasi-…
The notion of trivial extension of a ring by a module has been extensively studied and used in ring theory as well as in various other areas of research like cohomology theory, representation theory, category theory and homological algebra. In this paper we extend this classical ring construction by associating a ring to a ring $R$ and a family $M=(M_i)_{i=1}^{n}$ of $n$ $R$-modules for a given integer $n\geq 1$. We call this new ring construction an $n$-trivial extension of $R$ by $M$. In particular, the classical trivial extension will be just the $1$-trivial extension. Thus we generalize se…
Let $R$ be a ring satisfying a polynomial identity and let $D$ be a derivation of $R$. We consider the Jacobson radical of the skew polynomial ring $R[x;D]$ with coefficients in $R$ and with respect to $D$, and show that $J(R[x;D])\cap R$ is a nil $D$-ideal. This extends a result of Ferrero, Kishimoto, and Motose, who proved this in the case when $R$ is commutative.
In this paper we are concerned with the finiteness property of Ext-indices of several ring extensions. In this direction, we introduce some conjectures and discuss the relationship of them. Also we give affirmative answers to these conjectures in some special cases. Furthermore, we prove that the trivial extension of an Artinian local ring by its residue class field is always of finite Ext-index and we show that the Auslander-Reiten conjecture is true for this type of rings.
We give sufficient conditions for the descent construction to be the universal central extension of a twisted form of a split simple Lie algebra over a ring. In particular, the universal central extensions of twisted multiloop Lie tori are obtained by the descent construction.
Let A be a commutative ring and E a non-zero A-module. Necessary and sufficient conditions are given for the trivial ring extension R of A by E to be either arithmetical or Gaussian. The possibility for R to be B{é}zout is also studied, but a response is only given in the case where pSpec A (a quotient space of Spec} A) is totally disconnected. Trivial ring extensions which are fqp-rings are characterized only in the local case. To get a general result we intoduce the class of fqf-rings satisfying a weaker property than fqp-ring. Moreover, it is proven that the finitistic weak dimension of a f…
This paper partly settles a conjecture of Costa on (n,d)-rings, i.e., rings in which n-presented modules have projective dimension at most d. For this purpose, a theorem studies the transfer of the (n,d)-property to trivial extensions of local rings by their residue fields. It concludes with a brief discussion -backed by original examples- of the scopes and limits of our results.
A finite ring R and a weight w on R satisfy the Extension Property if every R-linear w-isometry between two R-linear codes in R^n extends to a monomial transformation of R^n that preserves w. MacWilliams proved that finite fields with the Hamming weight satisfy the Extension Property. It is known that finite Frobenius rings with either the Hamming weight or the homogeneous weight satisfy the Extension Property. Conversely, if a finite ring with the Hamming or homogeneous weight satisfies the Extension Property, then the ring is Frobenius. This paper addresses the question of a characterization…
We study the ideal theory of amalgamated products of Ore and quadratic extensions over a base ring R. We prove an analogue of the Hilbert Basis theorem for an amalgamated product Q of quadratic extensions and determine conditions for when the one-sided ideals of Q are principal or doubly-generated. We also determine conditions that make Q a principal ideal ring. Finally, we show that the double affine Hecke algebra $H_{q,t}$ associated to the general linear group GL_2(k) (here, k is a field with characteristic not 2) is an amalgamated product of quadratic extensions over a three-dimensional qu…
In this paper, our objects of interest are Hopf Galois extensions (e.g., Hopf algebras, Galois field extensions, strongly graded algebras, crossed products, principal bundles, etc.) and families of noncommutative rings (e.g., skew polynomial rings, PBW extensions and skew PBW extensions, etc.). We collect and systematize questions, problems, properties and recent advances in both theories by explicitly developing examples and doing calculations that are usually omitted in the literature. In particular, for Hopf Galois extensions we consider approaches from the point of view of quantum torsors …
A ring R shall be called F-noetherian if every finite subset of R is contained in a (left and right) noetherian subring of R . For example, every commutative ring is tightly F-noetherian in the sense that every finite subset of R generates a noetherian subring of R . F-noetherian rings have many interesting linear algebra properties which we refer to as the full strong rank condition, fully stably finite, and more generally the basic condition. We also study some basic ring-theoretic properties of F-noetherian rings such as localizations of F-noetherian rings. The F-noetherian property is pres…
We attach to any commutative ring R a subgroup of the Brauer group of R, called the Brauer-Galois group of R. Its elements are the classes of the Azumaya R-algebras which can be represented, via Brauer equivalence, by a Galois extension of R. We compute this group for some particular commutative fields.
We introduce the notion of totally reflexive extension of rings. It unifies Gorenstein orders and Frobenius extensions. We prove that for a totally reflexive extension, a module over the extension ring is totally reflexive if and only if its underlying module over the base ring is totally reflexive.
In this paper a framework to study the dual of skew cyclic codes is proposed. The transposed Hamming ring extensions are based in the existence of an anti-isomorphism of algebras between skew polynomial rings. Our construction is applied to left ideal convolutional codes, skew constacyclic codes and skew Reed-Solomon code, showing that the dual of these codes belong to the same class.
We give a sufficient condition for a bi-invariant weight on a Frobenius bimodule to satisfy the extension property. This condition applies to bi-invariant weights on a finite Frobenius ring as a special case. The complex-valued functions on a Frobenius bimodule are viewed as a module over the semigroup ring of the multiplicative semigroup of the coefficient ring.
In this paper we expand on previous results, studying the extent to which one can detect fusion in certain finite groups $Γ$, from information about the universal deformation rings of absolutely irreducible $\mathbb{F}_pΓ$-modules. We consider groups which are extensions of finite irreducible subgroups of $Gl_2({\mathbb{C}})$ by elementary abelian $p$-groups of rank $2$.
We apply set-theoretic methods to study projective modules and their generalizations over transfinite extensions of simple artinian rings R. We prove that if R is small, then the Weak Diamond implies that projectivity of an arbitrary module can be tested at the layer epimorphisms of R.
The MacWilliams extension theorem is investigated for various weight functions over finite Frobenius rings. The problem is reformulated in terms of a local-global property for subgroups of the general linear group. Among other things, it is shown that the extension theorem holds true for poset weights if and only if the underlying poset is hierarchical. Specifically, the Rosenbloom-Tsfasman weight for vector codes satisfies the extension theorem, whereas the Niederreiter-Rosenbloom-Tsfasman weight for matrix codes does not. A short character-theoretic proof of the well-known MacWilliams extens…
