Maximal Quotient Ring Research Articles

Equivalence between invariance conjectures for parabolic Kazhdan-Lusztig polynomials

We prove that the combinatorial invariance conjecture for parabolic Kazhdan-Lusztig polynomials, formulated by Mario Marietti, is equivalent to its restriction to maximal quotients. This equivalence lies at the other extreme in respect to the equivalence, recently proved by Barkley and Gaetz, with the invariance conjecture for Kazhdan-Lusztig polynomials, which turns out to be equivalent to the conjecture for maximal quotients.

Tautological characteristic classes II: the Witt class

AbstractLet K be an arbitrary infinite field. The cohomology group $$H^{2}(SL(2,K);H_2 SL(2,K))$$ H 2 ( S L ( 2 , K ) ; H 2 S L ( 2 , K ) ) contains the class of the universal central extension. When studying representations of fundamental groups of surfaces in SL(2, K) it is useful to have classes stable under deformations (Fenchel-Nielsen twists) of representations. We identify the maximal quotient of the universal class which is stable under twists as the Witt class of Nekovàř. The Milnor-Wood inequality asserts that an $$SL(2,\textbf{R})$$ S L ( 2 , R ) -bundle over a surface of genus g admits a flat structure if and only if its Euler number is $$\le (g - 1)$$ ≤ ( g - 1 ) . We establish an analog of this inequality, and a saturation result for the Witt class. The result is sharp for the field of rationals, but not sharp in general.

The lattice coordinatized by a ring

A coordinatization functor L ( − , 1 ) : Ring → Latt is defined from the category of rings to the category of modular lattices. The main features of this coordinatization functor are 1) that it extends the functor R ↦ L ( R ) of von Neumann that associates to a regular ring its lattice of principal right ideals; 2) it respects the respective ( − ) op endofunctors on Ring and Latt ; and 3) it admits localization at a left R -module. The complemented elements of L ( R , 1 ) form a partially ordered set S ( R ) isomorphic to the space of direct summands of R R . The right nonsingular rings for which the embedding of S ( R ) into the localization L ( R , 1 ) Q at the right maximal ring of quotients Q R is an isomorphism are characterized by a property that every finite matrix subgroup φ ( R R ) of the left R -module R R is essential in an element of S ( R ) . In that case, the space S ( R ) obtains the structure of a complemented modular lattice coordinatized by the dominion, or equivalently, the ring of definable scalars, of the maximal ring of quotients. The class of rings with this property is elementary, in contrast to the class of rings whose space of right summands is coordinatized by the maximal ring of quotients.

On the matrix wreath product of algebras

We study the singular ideal, the Jacobson radical and the maximal quotient rings of the matrix wreath product of two algebras and certain subrings as defined in (Alamhadi-Jain-Alsulami and Zelmanov, Electron. Res. Announc. Math. Sci. (2017)) [1]. We give formulas for computing the singular ideal and the Jacobson radical of a matrix wreath product. We also give the structure of the maximal right quotient ring of the infinite matrix ring of a right nonsingular algebra and, as consequence, we give the maximal quotient ring of a matrix wreath product. As an application of the fact that matrix wreath product of nonsingular algebras is nonsingular we construct another algebra (similar to the one in [3]) showing that any nonsingular algebra of countable dimension can be embedded in a finitely generated nonsingular algebra.

ON THE m_c-TOPOLOGY ON THE FUNCTIONALLY COUNTABLE SUBALGEBRA OF C(X)

In this paper, we consider the $m_c$-topology on $C_c(X)$, the functionally countable subalgebra of $C(X)$. We show that a Tychonoff space $X$ is countably pseudocompact if and only if the $m_c$-topology and the $u_c$-topology on $C_c(X)$ coincide. It is shown that whenever $X$ is a zero-dimensional space, then $C_c(X)$ is first countable if and only if $C(X)$ with the $m$-topology is first countable. Also, the set of all zero-divisors of $C_c(X)$ is closed if and only if $X$ is an almost $P$-space. We show that if $X$ is a strongly zero-dimensional space and $U$ is the set of all units of $C_c(X)$, then the maximal ring of quotients of $C_c(U)$ and $C_c(C_c(X))$ are isomorphic.

Rings of quotients of the ring consisting of ordered field valued continuous functions with countable range

For a zero-dimensional topological space X and a totally ordered field F with interval topology, Cc(X, F) denotes the ring consisting of ordered field-valued continuous functions with countable range on X. This article aims to study and investigate the rings of quotients of Cc(X,F). Qc(X,F) (resp. qc(X,F)), the maximal (resp. classical) ring of quotients of Cc(X,F) as a modified countable analogue of Q(X) (resp. q(X)), the maximal (resp. classical) ring of quotients of C(X) are characterized. It is proved that Qc(S), the maximal ring of quotients of the subring S of Cc(X,F), is a subring of Qc(X, F) if and only if every dense ideal in S has dense cozero-set in X. Also, the coincidence of rings of quotients of Cc(X, F) is investigated. We show that qc(X,F)=Cc(X, F) if and only if the set of non-units and zero-divisors in Cc(X,F) coincide if and only if X is almost CPF-space. Finally, it is shown that the fixed ring of quotients and the cofinite ring of quotients of Cc(X) coincide if and only if Hom(Mcp) = Cc(Xp) for every p ? X.

Ramification filtration via deformations

Let be a field of formal Laurent series with coefficients in a finite field of characteristic , the maximal quotient of the Galois group of of period and nilpotency class and the filtration by ramification subgroups in the upper numbering. Let be the identification of nilpotent Artin-Schreier theory: here is the group obtained from a suitable profinite Lie -algebra via the Campbell-Hausdorff composition law. We develop a new technique for describing the ideals such that and constructing their generators explicitly. Given , we construct an epimorphism of Lie algebras and an action of the formal group of order , , , on . Suppose , where , and is the ideal of generated by the elements of . The main result in the paper states that . In the last sections we relate this result to the explicit construction of generators of obtained previously by the author, develop a more efficient version of it and apply it to recover the whole ramification filtration of from the set of its jumps. Bibliography: 13 titles.

On the structure of generalized Jordan *-derivations of prime rings

Let be a noncommutative prime ring with involution and let be the maximal symmetric ring of quotients of In the present paper, we describe the structure of generalized Jordan *-derivations, i.e., additive mappings satisfying for all where d is an associated Jordan *-derivation of Precisely, we prove that under certain conditions any generalized Jordan *-derivation of is of the form for all where As a consequence, we show that any generalized Jordan *-derivation on a prime ring with involution is generalized X-inner, provided is not a PI-ring. Finally, we conclude our paper with a direction for further research.

Socle-injective semiprime rings, with some applications to Leavitt path algebras

A ring R is called right (or left) socle-injective if every R-homomorphism from the right (or left) socle of R into R extends to R. In this article, we show that any semiprime ring R with socle S, is socle-injective if and only if where Q′ is a suitable subring of maximal right ring of quotients of R and is an ideal of Q′. Furthermore, an explicit structure of the ring Q′ is presented for a semiprime socle- injective ring, with essential socle. As an application, we show that a unital Leavitt path algebra with essential socle is socle-injective if and only if is semisimple, hence von Neumann regular. Moreover, we observed that socle-injective Leavitt path algebras are left-right symmetric. We also have provided examples to illustrate our results.

An Engel condition with two generalized derivations in prime rings

Let R be a noncommutative prime ring with extended centroid C and maximal right ring of quotients Q. Let I be a nonzero ideal of R, and let g, h be two generalized derivations of R. Suppose that for all where are fixed positive integers. Then, one of the following conditions holds: there exist such that and for all m = s, n = t and there exist and such that g(x) = xa and for all C is a finite field, the matrix ring over C for some integer and there exist such that g(x) = xa and h(x) = ux for all

Weak Jordan derivations of prime rings

Let R be a prime ring with extended centroid C and with maximal left ring of quotients . An additive map is called a weak Jordan derivation if for all . Applying the theory of functional identities and dealing with the low dimensional cases, we give a complete characterization of weak Jordan derivations of prime rings. Moreover, we generalize Brešar's theorem concerning additive maps satisfying for all .

Leibniz algebras with low-dimensional maximal Lie quotients

Every Leibniz algebra has a maximal homomorphic image that is a Lie algebra. We classify cyclic Leibniz algebras over an arbitrary field. Such algebras have the 1-dimensional abelian Lie algebra as their maximal Lie quotient. We then give examples of Leibniz algebras whose associated maximal Lie quotients exhaust all 2-dimensional possibilities.

Goldie Rings Graded by a Group with Periodic Quotient Group Modulo the Center

In this paper, we study gr-prime and gr-semiprime Goldie rings graded by a group with periodic quotient group modulo the center. We enhance the theorem of Goodearl and Stafford (2000) about gr-prime rings graded by Abelian groups; we extend the Abelian group class to the class of groups with periodic quotient group modulo the center. We also decompose the orthogonal graded completion Ogr(R) of a gr-semiprime Goldie ring R (graded by a group satisfying the same condition) into a direct sum of gr-prime Goldie rings R1, . . . , Rn and prove that the maximal graded quotient ring Qgr(R) equals the direct sum of classical graded quotients rings of R1, . . . , Rn.

Certain basic functional identities of semiprime rings

The goal of the article is to discuss certain basic functional identities of semiprime rings. Specifically, the following result is proved: Let R be a semiprime ring with extended centroid C and let be additive maps, where denotes the maximal left ring of quotients of R. Suppose that for all . Then there exist and additive maps such that and for all . Moreover, some related functional identities are also characterized.

Centralizers and Jordan triple derivations of semiprime rings

Let R be a semiprime ring with extended centroid C and with maximal left ring of quotients . An additive map is called a Jordan triple derivation if for all . In 1957, Herstein proved that a Jordan triple derivation, which is also a Jordan derivation, of a noncommutative prime ring of characteristic 2, must be a derivation. In 1989, Brešar proved that any Jordan triple derivation of a 2-torsion free semiprime ring is a derivation. In the article, we give a complete characterization of Jordan triple derivations of arbitrary semiprime rings. To get such a characterization we first show that, in some sense, an additive map satisfying for all can be realized as a centralizer with only an exceptional case that and R is commutative.

Groups of automorphisms of local fields of period p and nilpotent class < p, II

Suppose [Formula: see text] is a finite field extension of [Formula: see text] containing a primitive [Formula: see text]th root of unity. Let [Formula: see text] be the maximal quotient of period [Formula: see text] and nilpotent class [Formula: see text] of the Galois group of a maximal [Formula: see text]-extension of [Formula: see text]. We describe the ramification filtration [Formula: see text] and relate it to an explicit form of the Demushkin relation for [Formula: see text]. The results are given in terms of Lie algebras attached to the appropriate [Formula: see text]-groups by the classical equivalence of the categories of [Formula: see text]-groups and Lie algebras of nilpotent class [Formula: see text].

Generalized skew derivations with centralizer conditions on multilinear polynomials

Let R be a noncommutative prime ring of characteristic not 2 with extended centroid C, the maximal right ring of quotients Q and a nonzero generalized skew derivation \(\delta \). Assume that \(f(X_{1},\ldots ,X_{n})\) is a multilinear polynomial over C that is not central-valued on R and f(R) is the set of all evaluations of the multilinear polynomial \(f\big (X_{1},\ldots ,X_{n}\big )\) in R. Denote the set \(S:= \big \{\delta (u)u \mid u \in f(R) \big \}\). The goal of the paper is to study \(C_R(S)\), the centralizer of S in R. To be precise, given a noncentral element \(b\in R\) it is proved that if \(b\in C_R(S)\), i.e., $$\begin{aligned}{}[\delta (u)u, b]=0 \end{aligned}$$ for all \(u \in f(R)\), then there exists \(a \in Q\) with \([a, b]=0\) such that \(\delta (x)=ax\) for all \(x\in R\) and \(f\big (X_{1},\ldots ,X_{n}\big )^{2}\) is central-valued on R. As applications to the theorem, we consider the case of \(\delta (u)u \in C\) for all \(u \in f(R)\) and also we investigate commuting values of two generalized skew derivations having different associated skew derivations on multilinear polynomials.

Jordan [! \large !]$\sigma $-derivations of prime rings

Let $R$ be a noncommutative prime ring with extended centroid~$C$ and with $Q_{mr}(R)$ its maximal right ring of quotients. From the viewpoint of functional identities, we give a complete characterization of Jordan $\sigma $-derivations of $R$ with $\sigma $ an epimorphism. Precisely, given such a Jordan $\sigma $-derivation $\de \colon R\to Q_{mr}(R)$, it is proved that either $\delta $ is a $\sigma $-derivation or a derivation $d\colon R\to Q_{mr}(R)$ and a unit $u\in Q_{mr}(R)$ exist such that $\delta (x)=ud(x)+\mu (x)u$ for all $x\in R$, where $\mu \colon R\to C$ is an additive map satisfying $\mu (x^2)=0$ for all $x\in R$. In addition, if $\sigma $ is an X-outer automorphism, then $\delta $ is always a $\sigma $-derivation.

Rationality of homogeneous varieties

Let G be a connected linear algebraic group over an algebraically closed field k, and let H be a connected closed subgroup of G. We prove that the homogeneous variety G/H is a rational variety over k whenever H is solvable, or when dim(G/H) < 11 and characteristic(k) = 0. When H is of maximal rank in G, we also prove that G/H is rational if the maximal semisimple quotient of G is isogenous to a product of almost-simple groups of type A, type C (when characteristic(k) is not 2), or type B_3 or G_2 (when characteristic(k) = 0).

Biderivations of triangular rings

In this paper, we prove that every biderivation of certain triangular rings is the sum of an extremal biderivation and an inner biderivation, using the notion of maximal left ring of quotients. As a consequence, we show that every biderivation of the ring of all upper triangular matrices over a unital ring () is the sum of an extremal biderivation and an inner biderivation, which extends two results of Benkovič and Ghosseiri, respectively.