Quotient Ring Research Articles

Quotient rings satisfying some identities

This paper investigates the commutativity of the quotient ring \(\mathcal{R}/P\), where \(\mathcal{R}\) is an associative ring with a prime ideal \(P\), and the possibility of forms of derivations satisfying certain algebraic identities on \(\mathcal{R}\). We provide some results for strong commutativity-preserving derivations of prime rings.

Equivariant resolutions over Veronese rings

AbstractWorking in a polynomial ring , where is an arbitrary commutative ring with 1, we consider the th Veronese subalgebras , as well as natural ‐submodules inside . We develop and use characteristic‐free theory of Schur functors associated to ribbon skew diagrams as a tool to construct simple ‐equivariant minimal free ‐resolutions for the quotient ring and for these modules . These also lead to elegant descriptions of for all and for any pair of these modules .

Representation theory and the diagonal coinvariant ring of the type B Weyl group

We explain how to use representation theory to give a lower bound on the dimension of the quotient ring by type Bn diagonal invariants that improves upon the current known lower bound (2n+1)n by a quadratic polynomial in n.

Cohomology of smooth toric varieties: Naturality

Building on the recent computation of the cohomology rings of smooth toric varieties and partial quotients of moment-angle complexes, we investigate the naturality properties of the resulting isomorphism between the cohomology of such a space and the torsion product involving the Stanley–Reisner ring. If 2 is invertible in the chosen coefficient ring, then the isomorphism is natural with respect to toric morphisms, which for partial quotients are defined in analogy with toric varieties. In general there are deformation terms that we describe explicitly.

Product of generalized derivations with Engel condition on Lie ideals in prime rings

This paper represents the structures of generalized derivations satisfying an identity with [Formula: see text]-Engel condition. To prove our result, we take [Formula: see text] as a prime ring with [Formula: see text], and [Formula: see text] as Utumi quotient ring of [Formula: see text]. The center of [Formula: see text] is [Formula: see text] which is called extended centroid of [Formula: see text]. Let [Formula: see text] be a non-central Lie ideal of [Formula: see text]. Suppose that [Formula: see text] and [Formula: see text] are two nonzero generalized derivations of [Formula: see text] and [Formula: see text], any fixed integer. If [Formula: see text] for all [Formula: see text], then one of the below-mentioned conclusions holds: (1) [Formula: see text] and [Formula: see text] for all [Formula: see text] and for some [Formula: see text] with [Formula: see text]; (2) [Formula: see text] satisfies [Formula: see text].

On the hulls of cyclic codes of oddly even length over [formula omitted

In this paper, we study hulls of cyclic codes of length 2n (i.e., oddly even length) over the ring Z4 of integers modulo 4 by viewing these codes as ideals of the quotient ring Z4[x]/<x2n−1>, where n is an odd positive integer. We express the generator polynomials of the hull of each cyclic code C of length 2n over Z4 in terms of the generator polynomials of the code C, and we note that the 2-dimension of the hull of the code C is an integer Θ satisfying 0≤Θ≤2n. We also obtain an enumeration formula for cyclic codes of length 2n over Z4 with hulls of a given 2-dimension. Besides this, we study the average 2-dimension, denoted by E(2n), of the hulls of cyclic codes of length 2n over Z4 and establish a formula for E(2n) that handles well. With the help of this formula, we deduce that E(2n)=5n6 when n∈N2, and we study the growth rate of E(2n) with respect to the length 2n when n∉N2, where N2 denotes the set of all positive integers ω such that w divides 2i+1 for some positive integer i. Further, when n∉N2, we derive lower and upper bounds on E(2n) and show that these bounds are attained at n=7. We also illustrate these results with some examples. As an application of these results, we construct some entanglement-assisted quantum error-correcting codes (EAQECCs) over Z4 with specific parameters from cyclic codes of oddly even length over Z4.

Depth and Stanley Depth of the Edge Ideals of r-Fold Bristled Graphs of Some Graphs

In this paper, we find values of depth, Stanley depth, and projective dimension of the quotient rings of the edge ideals associated with r-fold bristled graphs of ladder graphs, circular ladder graphs, some king’s graphs, and circular king’s graphs.

On graded (1,r)-ideals

Let [Formula: see text] be a group with identity element [Formula: see text] and [Formula: see text] be a commutative [Formula: see text]-graded ring with nonzero unity [Formula: see text]. In this paper, we introduce the graded version of [Formula: see text]-ideals which is a generalization of graded [Formula: see text]-ideals. A proper graded ideal [Formula: see text] of [Formula: see text] is said to be a graded [Formula: see text]-ideal if whenever [Formula: see text] for some nonunits homogeneous elements [Formula: see text], then either [Formula: see text] or [Formula: see text]. We investigate some basic properties of graded [Formula: see text]-ideals. We show that if [Formula: see text] admits a graded [Formula: see text]-ideal that is not a graded [Formula: see text]-ideal, then [Formula: see text] is a [Formula: see text]-graded local ring. Also, we give a method to construct graded [Formula: see text]-ideals that are not graded [Formula: see text]-ideals. Furthermore, we prove that [Formula: see text] is a graded total quotient ring if and only if every proper graded ideal of [Formula: see text] is graded [Formula: see text]-ideal and also we present a counterpart of prime avoidance lemma for graded [Formula: see text]-ideals. Finally, an idea is given about some graded [Formula: see text]-ideals of the ring of fractions and the idealization.

Generalized skew derivations as a generalization of left Jordan multipliers in prime rings

Let m, n be two nonzero fixed positive integers, R a prime ring with the right Martindale quotient ring Q, G and H two generalized skew derivations of R with the same associated automorphism α of R. If G ( x m + n ) = H ( x m ) x n is an identity for R, then either R is commutative or there exists q ∈ Q such that G ( x ) = H ( x ) = q x , for any x ∈ R .

Actions of generalized derivations on prime ideals in $*$-rings with applications

In this paper, we make use of generalized derivations to scrutinize the deportment of prime ideal satisfying certain algebraic $*$-identities in rings with involution. In specific cases, the structure of the quotient ring $\mathscr{R}/\mathscr{P}$ will be resolved, where $\mathscr{R}$ is an arbitrary ring and $\mathscr{P}$ is a prime ideal of $\mathscr{R}$ and we also find the behaviour of derivations associated with generalized derivations satisfying algebraic $*$-identities involving prime ideals. Finally, we conclude our paper with applications of the previous section's results.

Generalized Reflexive Structures Properties of Crossed Products Type

Let $R$ be a ring and $M$ be a monoid with a twisting map $f : M \times M \rightarrow U(R)$ and an action map $\omega : M \rightarrow Aut(R)$. The objective of our work is to extend the reflexive properties of rings by focusing on the crossed product $R \ast M$ over $R$. In order to achieve this, we introduce and examine the concept of strongly $CM$-reflexive rings. Although a monoid $M$ and any ring $R$ with an idempotent are not strongly $CM$-reflexive in general, we prove that $R$ is strongly $CM$-reflexive under some additional conditions. Moreover, we prove that if $R$ is a left $p.q.$-Baer (semiprime, left $APP$-ring, respectively), then $R$ is strongly $CM$-reflexive. Additionally, for a right Ore ring $R$ with a classical right quotient ring $Q$, we prove $R$ is strongly $CM$-reflexive if and only if $Q$ is strongly $CM$-reflexive. Finally, we discuss some relevant results on crossed products.

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.

On Symmetric Additive Mappings and Their Applications

The key motive of this paper is to study symmetric additive mappings and discuss their applications. The study of these symmetric mappings makes it possible to characterize symmetric n-derivations and describe the structure of the quotient ring S/P, where S is any ring and P is a prime ideal of S. The symmetricity of additive mappings allows us to transfer ring theory results to functional analyses, particularly to C∗-algebras. Precisely, we describe the structures of C∗-algebras via symmetric additive mappings.

Relatively polyform modules

In this paper, we investigate polyform modules and introduce the notion of relatively polyform modules over a ring [Formula: see text]. Some new characterizations and properties of polyform modules are obtained. For a retractable polyform module [Formula: see text], it is proved that the maximal right ring of quotients of End[Formula: see text] is End[Formula: see text], where [Formula: see text] is the rational hull of [Formula: see text]. Moreover, we characterize nonsingular modules by using the relatively polyform property. In addition, it is proved that the class of all polyform right [Formula: see text]-modules is closed under submodules and rational extensions.

Units and linear independence

An algebra [Formula: see text] will be called fluid if for any linearly independent set [Formula: see text] consisting of units, the set [Formula: see text] of inverses of the elements of [Formula: see text] is also linearly independent. We consider various examples of fluid algebras and study fluid algebras in some specific settings including direct sums, field extensions, quotient rings of the ring of polynomials over a field, and matrix algebras. Pertinent parameters for the study of fluidity are also introduced and studied.

Jordan homoderivation behavior of generalized derivations in prime rings

UDC 512.5 Suppose that R is a prime ring with c h a r ( R ) ≠ 2 and f ( ξ 1 , … , ξ n ) is a noncentral multilinear polynomial over C ( = Z ( U ) ) , where U is the Utumi quotient ring of R . An additive mapping h : R → R is called homoderivation if h ( a b ) = h ( a ) h ( b ) + h ( a ) b + a h ( b ) for all a , b ∈ R . We investigate the behavior of three generalized derivations F , G , and H of R satisfying the condition F ( ξ 2 ) = G ( ξ ) 2 + H ( ξ ) ξ + ξ H ( ξ ) for all ξ inf ( R ) = { f ( ξ 1 , … , ξ n ) ∣ ξ 1 , … , ξ n ∈ R } .

Annihilating and co-commuting conditions with generalized skew derivations

Let R be a prime ring of characteristic different from 2, Qr its right Martindale quotient ring and C its extended centroid. Suppose that F, G are generalized skew derivations of R associated with the same automorphism α of R, I a non-central ideal of R and 0 ≠ a ∈ R . If a ( F ( x ) x − x G ( x ) ) = 0 , for all x ∈ I , then there exist b , b ′ , q ∈ Q r and λ , ϑ ∈ C such that F ( x ) = b x + qxb ′ , G ( x ) = ( ϑ b ′ + λ ) x , for any x ∈ R , with a ( b − λ ) = 0 and a ( q − ϑ ) = 0 . In particular, if α is not an inner automorphism of R, then F(x) = bx, G ( x ) = λ x , for any x ∈ R , with a ( b − λ ) = 0 .

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.

Nonlinear bi-skew Jordan derivations in prime ∗-rings

Let [Formula: see text] be a unital prime ∗-ring containing a nontrivial symmetric idempotent and let [Formula: see text] be the maximal symmetric ring of quotients of [Formula: see text]. Using the technique of Peirce decomposition and the theory of functional identities, we prove that a map [Formula: see text] satisfies [Formula: see text] for all [Formula: see text] if and only if [Formula: see text] is an additive ∗-derivation unless [Formula: see text] and [Formula: see text]. As an application, we shall characterize such maps in different operator algebras.

Derivations characterized by monomials x2n in prime rings

Let [Formula: see text] be a prime ring of [Formula: see text] or [Formula: see text] and let [Formula: see text] be an additive map such that [Formula: see text] for all [Formula: see text], where [Formula: see text] is a positive integer and [Formula: see text] is the maximal symmetric ring of quotients of [Formula: see text]. It is shown that there exist a derivation [Formula: see text] and an additive map [Formula: see text] with [Formula: see text] for all [Formula: see text], such that [Formula: see text]. This result is a natural generalization of the classic theorem of Herstein for Jordan derivations on prime rings. Moreover, it gives a purely algebraic version of the theorem recently obtained by Kosi-Ulbl, Rodriguez and Vukman for standard operator algebras on Banach spaces.