Left Quotient Ring Research Articles

Characterization of n-Lie-type derivations on triangular rings

By means of the maximal left ring of quotients, this paper proves that every n-Lie m-derivation on a triangular ring can be resolved into a sum of an extremal n-derivation and an n-additive central mapping. As applications, n-Lie m-derivations on some classical triangular rings are characterized.

Generalized Lie n-derivations on arbitrary triangular algebras

Abstract In this study, we consider generalized Lie n n -derivations of an arbitrary triangular algebra T T through the constructed triangular algebra T 0 {T}_{0} , where T 0 {T}_{0} is constructed using the notion of maximal left (right) ring of quotients.

X-generalized skew derivations behaving as weak Jordan right η-centralizers in prime rings

Consider a prime ring R which is noncommutative with the maximal left ring of quotients of R denoted by Q m l ( R ) and the extended centroid C . An additive map G : R → Q m l ( R ) satisfying G ( g 2 ) + η g G ( g ) ∈ C for all g ∈ R , where η ∈ C is called a weak Jordan right η-centralizer. In this paper, we establish the characterization of weak Jordan right η-centralizers in prime rings. As an application, we describe X-generalized skew derivations which behave like weak Jordan right η-centralizers.

Bi-Lie n-derivations on triangular rings

<abstract><p>The purpose of this article is to prove that every bi-Lie n-derivation of certain triangular rings is the sum of an inner biderivation, an extremal biderivation and an additive central mapping vanishing at $ (n-1)^{th} $-commutators for both components, using the notion of maximal left ring of quotients. As a consequence, we characterize the decomposition structure of bi-Lie n-derivations on upper triangular matrix rings.</p></abstract>

Functional identities of degree 2 vanishing on zero products of XY and Y X

Let R be a unitary prime ring with the maximal left ring of quotients . We completely characterize the forms of additive maps such that whenever satisfy xy = 0 = yx under the assumptions that R contains a nontrivial idempotent and characteristic of R is not 2. This partially generalizes a result of T.K. Lee in [Bi-additive maps of ζ-Lie product type vanishing on zero products of XY and Y X. Comm Algebra. 2017;45(8):3449–3467].

Engel conditions of generalized derivations on left ideals and Lie ideals in prime rings

Let R be a noncommutative prime ring, I a nonzero left ideal of R, L a non-central Lie ideal of R, U the left Utumi quotient ring of R and the extended centroid of R. Let G be a generalized derivation of R and g be a derivation of R and fixed positive integers. In the present paper, we describe the structure of R and all possible forms of G and g in the following cases: for all and 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 .

Rings of invariants of finite groups when the bad primes exist

Let R be a ring (not necessarily with 1) and G be a finite group of automorphisms of R. The set of primes p such that and R is not p-torsion free, is called the set of bad primes. When the ring is -torsion free, i.e. the properties of the rings R and RG are closely connected. The aim of the article is to show that this is also true when under natural conditions on bad primes. In particular, it is shown that the Jacobson radical (respectively, the prime radical) of the ring RG is equal to the intersection of the Jacobson radical (respectively, the prime radical) of R with RG; if the ring R is semiprime then so is RG; if the trace of the ring R is nilpotent then the ring itself is nilpotent; if R is a semiprime ring then R is left Goldie iff the ring RG is so, and in this case, the ring of G-invariants of the left quotient ring of R is isomorphic to the left quotient ring of RG and

Lie (Jordan) derivations of arbitrary triangular algebras

In this paper we construct a triangular algebra from a given triangular algebra, using the notion of maximal left (right) ring of quotients. As an application we give a description of Lie (Jordan) derivations of arbitrary triangular algebras through the constructed triangular algebra.

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.

Left localizable rings and their characterizations

A new class of rings, the class of left localizable rings, is introduced. A ring R is left localizable if each nonzero element of R is invertible in some left localization S−1R of the ring R. Explicit criteria are given for a ring to be a left localizable ring provided the ring has only finitely many maximal left denominator sets (e.g., this is the case if a ring has a left Noetherian left quotient ring). It is proved that a ring with finitely many maximal left denominator sets is a left localizable ring iff its left quotient ring is a direct product of finitely many division rings. A characterization is given of the class of rings that are finite direct product of left localization maximal rings.

Criteria for a ring to have a left Noetherian left quotient ring

Two criteria are given for a ring to have a left Noetherian left quotient ring (to find a criterion was an open problem since 70's). It is proved that each such ring has only finitely many maximal left denominator sets.

Criteria for a Ring to have aLeft Noetherian Largest Left Quotient Ring

Criteria are given for a ring to have a left Noetherian largest left quotient ring. It is proved that each such a ring has only finitely many maximal left denominator sets. An explicit description of them is given. In particular, every left Noetherian ring has only finitely many maximal left denominator sets.

The classical left regular left quotient ring of a ring and its semisimplicity criteria

Let [Formula: see text] be a ring, [Formula: see text] and [Formula: see text] be the set of regular and left regular elements of [Formula: see text] ([Formula: see text]). Goldie’s Theorem is a semisimplicity criterion for the classical left quotient ring [Formula: see text]. Semisimplicity criteria are given for the classical left regular left quotient ring [Formula: see text]. As a corollary, two new semisimplicity criteria for [Formula: see text] are obtained (in the spirit of Goldie).

Biderivations of Triangular Rings Revisited

We consider the problem of describing the form of biderivations of a triangular ring. Our approach is based on the notion of the maximal left ring of quotients, which enables us to generalize Benkovic’s result on biderivations (Benkovic in Linear Algebra Appl 431:1587–1602, 2009). Our result is applied to block upper triangular matrix rings and nest algebras.

Weakly left localizable rings

ABSTRACTA new class of rings, the class of weakly left localizable rings, is introduced. A ring R is called weakly left localizable if each non-nilpotent element of R is invertible in some left localization S−1R of the ring R. Explicit criteria are given for a ring to be a weakly left localizable ring provided the ring has only finitely many maximal left denominator sets (eg, this is the case for all left Noetherian rings). It is proved that a ring with finitely many maximal left denominator sets that satisfies some natural conditions is a weakly left localizable ring iff its left quotient ring is a direct product of finitely many local rings such that their radicals are nil ideals.

On a ring structure related to annihilators

In this note, we focus our attention on a new ring structure related to annihilators, and consider a ring property that contains many kinds of ring classes, introducing right ZAFS. This property is shown to be not left-right symmetric but left-right symmetric for left or right Artinian rings. The left (right) ZAFS property is shown to pass to Ore extensions with automorphisms. The left (respectively, right) ZAFS property is shown to pass also to classical left (respectively, right) quotient rings, yielding that semiprime right Goldie rings are ZAFS.

The Largest Left Quotient Ring of a Ring

The left quotient ring (i.e., the left classical ring of fractions) Qcl(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 Ql(R) of an arbitrary ring R is proved, i.e., Ql(R) = S0(R)−1R where S0(R) is the largest left regular denominator set of R. It is proved that Ql(Ql(R)) = Ql(R); the ring Ql(R) is semisimple iff Qcl(R) exists and is semisimple; moreover, if the ring Ql(R) is left Artinian, then Qcl(R) exists and Ql(R) = Qcl(R). The group of units Ql(R)* of Ql(R) is equal to the set {s−1t | s, t ∈ S0(R)} and S0(R) = R ∩ Ql(R)*. If there exists a finitely generated flat left R-module which is not projective, then Ql(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).

A Characterization of Generalized Derivations on Prime Rings

Let R be a noncommutative prime ring, U be the left Utumi quotient ring of R, and k, m, n, r be fixed positive integers. If there exist a generalized derivation G and a derivation g (which is independent of G) of R such that [G(xm)xn + xng(xm), xr]k = 0, for all x ∈ R, then there exists a ∈ U such that G(x) = ax, for all x ∈ R. As a consequence of the result in the present article, one may obtain Theorem 1 in Demir and Argaç [10].