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Let R be a prime ring of char R = 0 or char R > 2 n and let d : R → Q m s (R) be an additive map such that d (x 2 n) = d (x n) x n + x n d (x n) for all x ∈ R , where n is a positive integer and Q m s (R) is the maximal symmetric ring of quotients of R. It is shown that there exist a derivation δ : R → Q m s (R) and an additive map μ : R → Q m s (R) with μ (x n) = 0 for all x ∈ R , such that d = δ + μ. 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. [ABSTRACT FROM AUTHOR]

Let R be a commutative Noetherian ring, I an ideal of R and M a finitely generated R-module with dimR(M) = d. Denote by depthR(I,M) the depth of M in I. In [C. Huneke and V. Trivedi, The height of ideals and regular sequences, Manuscr. Math. 93 (1997) 137–142], Huneke and Trivedi proved that if R is a quotient of a regular ring then there exists a finite subset ΛM of Spec(R) such that depthR(I,M) =min픭∈Λ M{depthR픭(M픭) + ht((I + 픭)/픭)}. Denote by PsuppRi(M) = {픭 ∈Spec(R)|H픭 R픭i−dim(R/픭)(M픭)≠0} the ith pseudo support of M defined by Brodmann and Sharp [On the dimension and multiplicity of local cohomology modules, Nagoya Math. J. 167 (2002) 217–233]. In this paper, we prove that if PsuppRi(M) is closed for all i ≤ d then the above formula of depthR(I,M) holds true, where ΛM =⋃0≤i≤dmin PsuppRi(M). In particular, if R is a quotient of a Cohen–Macaulay local ring then ΛM = ⋃0≤i≤dminVar(AnnR(H픪i(M))). We also give some examples to clarify the results. [ABSTRACT FROM AUTHOR]

Our fundamental aim in this paper is to study certain * {*} -identities with pair of derivations acting on a prime ideal P and investigate commutativity of a quotient ring R / P {R/P} where R is any ring. Some well-known results characterizing commutativity of prime rings have been generalized. Finally, examples are given to demonstrate that the restrictions imposed on the hypothesis of our results are not superfluous. [ABSTRACT FROM AUTHOR]

For each parabolic subgroup \mathrm {P} of the general linear group \operatorname {GL}_n(\mathbb {F}_q), a conjecture due to Lewis, Reiner and Stanton [Proc. Roy. Soc. Edinburgh Sect. A 147 (2017), pp. 831–873] predicts a formula for the Hilbert series of the space of invariants \mathcal {Q}_m(n)^\mathrm {P} where \mathcal {Q}_m(n) is the quotient ring \mathbb {F}_q[x_1,\ldots,x_n]/(x_1^{q^m},\ldots,x_n^{q^m}). In this paper, we prove the conjecture for the Borel subgroup \mathrm {B} by constructing a linear basis for \mathcal {Q}_m(n)^\mathrm {B}. The construction is based on an operator \delta which produces new invariants from old invariants of lower ranks. We also upgrade the conjecture of Lewis, Reiner and Stanton by proposing an explicit basis for the space of invariants for each parabolic subgroup. [ABSTRACT FROM AUTHOR]

We introduce a presentation of the Chow ring of a matroid by a new set of generators, called "simplicial generators." These generators are analogous to nef divisors on projective toric varieties, and admit a combinatorial interpretation via the theory of matroid quotients. Using this combinatorial interpretation, we (i) produce a bijection between a monomial basis of the Chow ring and a relative generalization of Schubert matroids, (ii) recover the Poincaré duality property, (iii) give a formula for the volume polynomial, which we show is log-concave in the positive orthant, and (iv) recover the validity of Hodge-Riemann relations in degree 1, which is the part of the Hodge theory of matroids that currently accounts for all combinatorial applications of the work of Adiprasito et al. (2018). Our work avoids the use of "flips," the key technical tool employed in that work. [ABSTRACT FROM AUTHOR]

Let R symbolize a prime ring with a characteristic other than 2, and let ρ (η 1 , ... , η n) represent a non-central multilinear polynomial over the extended centroid C of R . Assuming the presence of three b-generalized skew derivations on R , namely D 1 , D 2 , and D 3 that satisfy the equation D 1 (ρ) ρ + ρ D 2 (ρ) = D 3 (ρ 2) for all ρ = ρ (η 1 , ... , η n) , where η 1 , ... , η n are elements of R . In this research paper, our goal is to elucidate all potential configurations of D 1 , D 2 , and D 3 . [ABSTRACT FROM AUTHOR]

A ring is called a right a-ring if every right ideal is automorphism invariant. We describe some properties of a-rings over semiperfect rings. It is shown that an I-finite right a-ring is a direct sum of a semisimple Artinian ring and a basic ring. It is also demonstrated that if R is an indecomposable (as a ring) I-finite right a-ring not simple with nontrivial idempotents such that every minimal right ideal is a right annihilator and Soc(RR) = Soc(RR) is essential in RR, then R is a quasi-Frobenius ring, which is also a right q-ring. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

This paper examines the commutativity of the quotient ring F / Y by utilizing specific differential identities in a general ring F that contains a semiprime ideal Y. This study particularly focuses on the role of a multiplicative generalized semiderivation ψ , which is associated with a map θ , in determining the commutative nature of the quotient ring. [ABSTRACT FROM AUTHOR]

We prove that cellular Noetherian algebras with finite global dimension are split quasi-hereditary over a regular commutative Noetherian ring with finite Krull dimension and their quasi-hereditary structure is unique, up to equivalence. In the process, we establish that a split quasi-hereditary algebra is semi-perfect if and only if the ground ring is a local commutative Noetherian ring. We give a formula to determine the global dimension of a split quasi-hereditary algebra over a commutative regular Noetherian ring (with finite Krull dimension) in terms of the ground ring and finite-dimensional split quasi-hereditary algebras. For the general case, we give upper bounds for the finitistic dimension of split quasi-hereditary algebras over arbitrary commutative Noetherian rings. We apply these results to Schur algebras over regular Noetherian rings and to Schur algebras over quotients rings of the integers. [ABSTRACT FROM AUTHOR]

Let F, G, and H be generalized derivations with F ≠ 0 on a prime ring R of characteristic different from 2, with Utumi quotient ring U and extended centroid C such that for all r ∈ R n FH (f (r) 2) = F (f (r)) G (f (r)) + G (f (r)) F (f (r)) (1) where f(r) is a multilinear polynomial over C which is noncentral valued on R. In this article we prove that if R does not satisfy the standard identity s4 then either R satisfies (1) or f (x 1 , ... , x n) 2 is central valued on R. Also in both the cases we describe all the possible forms of F, G, and H. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the notions of radical filters and extended filters of Intuitionistic Linear algebras (IL-algebras for short) and give some of their properties. The notion of closure operation on an IL-algebra is also introduced as well as the study of some of their main properties. The radical of filters and extended filters are examples of closure operations among several others provided. The class of stable closure operations on an IL-algebra L is used to study the unifying properties of some subclasses of the lattice of filters of L. In particular, we obtain that for a stable closure operation c on an IL-algebra, the collection of c-closed elements of its lattice of filters forms a complete Heyting algebra. Hyperarchimedean IL-algebras are also characterized using closure operations. It is shown that the image of a semi-prime closure operation on an IL-algebra is isomorphic to a quotient IL-algebra. Some properties of the quotients induced by closure operations on an IL-algebra are explored. [ABSTRACT FROM AUTHOR]

The purpose of this article is to enhance the previous studies regarding the behavior of a quotient ring ℑ / ℘ , where ℘ is a prime ideal in a ring ℑ. In particular, we are going to explore more general scenarios whenever a ring ℑ admits a generalized (α , β) -derivation associated with an (α , β) -derivation ∂ that satisfies certain criteria involving ℘, where α and β are automorphisms on ℑ. Moreover, we provide some examples to demonstrate the importance of the assumptions made in our results. [ABSTRACT FROM AUTHOR]

The existence of a maximal ideal in a general nontrivial commutative ring is tied together with the axiom of choice. Following Berardi, Valentini and thus Krivine but using the relative interpretation of negation (that is, as "implies 0 = 1 ") we show, in constructive set theory with minimal logic, how for countable rings one can do without any kind of choice and without the usual decidability assumption that the ring is strongly discrete (membership in finitely generated ideals is decidable). By a functional recursive definition we obtain a maximal ideal in the sense that the quotient ring is a residue field (every noninvertible element is zero), and with strong discreteness even a geometric field (every element is either invertible or else zero). Krull's lemma for the related notion of prime ideal follows by passing to rings of fractions. By employing a construction variant of set-theoretic forcing due to Joyal and Tierney, we expand our treatment to arbitrary rings and establish a connection with dynamical algebra: We recover the dynamical approach to maximal ideals as a parametrized version of the celebrated double negation translation. This connection allows us to give formal a priori criteria elucidating the scope of the dynamical method. Along the way we do a case study for proofs in algebra with minimal logic, and generalize the construction to arbitrary inconsistency predicates. A partial Agda formalization is available at an accompanying repository.1 See https://github.com/iblech/constructive-maximal-ideals/. This text is a revised and extended version of the conference paper (In Revolutions and Revelations in Computability. 18th Conference on Computability in Europe (2022) Springer). The conference paper only briefly sketched the connection with dynamical algebra; did not compare this connection with other flavors of set-theoretic forcing; and sticked to the case of commutative algebra only, passing on the generalization to inconsistency predicates and well-orders. [ABSTRACT FROM AUTHOR]

Let P (x) be a system of polynomials in s variables, where x ∈ C s . If z 0 is an isolated zero of P, then the multiplicity and its structure at z 0 can be revealed by the normal set of the quotient ring R (< P >) or its dual space R * or by certain numerical methods. In his book titled "Numerical Polynomial Algebra", Stetter described the so-called distinguished points, which are embedded in a zero manifold of P, and the author defined their multiplicities. In this note, we will generalize the definition of distinguished points and give a more appropriate definition for their multiplicity, as well as show how to calculate the multiplicity of these points. [ABSTRACT FROM AUTHOR]

An algebra A will be called fluid if for any linearly independent set X consisting of units, the set X − 1 of inverses of the elements of X 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. [ABSTRACT FROM AUTHOR]

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

In this paper, we study the structure of the skew polynomial ring R = ( p + u p) [ x ; ] and its quotient ring R n = R / 〈 x n − 1 〉 , where p is an odd prime number, u 2 = 0 , and (u) = − u. We give an explicit structure of the ideals in R and R n and propose an algorithm to characterize them. We identify the structure of prime, maximal, and primary ideals in these rings. In particular, we prove that this group ring is not Laskerian. [ABSTRACT FROM AUTHOR]

Let R be a noncommutative prime ring of characteristic different from 2 , with right Martindale quotient ring Q r and extended centroid C. Let m , n ≥ 1 be fixed integers and F a nonzero generalized skew derivation of R. In this paper, we investigate the set S = { F (x m) x n + x n F (x m) : x ∈ R } and prove that its left annihilator in R is identically zero. Using the above result, we also study the identity F (x) x n + x n F (x) = 0 for semiprime rings. [ABSTRACT FROM AUTHOR]

This paper explores the topological properties of irresolute topological groups, their quotient maps, and the role of topology in normal subgroups. It provides a detailed analysis using examples and counterexamples. The study focuses on the essential features of irresolute topological groups and their quotient groups, for understanding the topological aspects of isotropy groups. For a transformation group (H,Y, ψ) and a point y ∈ Y, the set Hy = {h ∈ H: hy = y} consisting of elements of H that fix y, is called the isotropy group at y. The paper highlights the distinct topological characteristics of isotropy groups in transformation group structure. It demonstrates that if (H,Y, ψ) is an Irr∗-topological transformation group, then (H/Ker ψ,Y, ψ) forms an effective Irr∗-topological transformation group. By investigating both irresolute topological groups and isotropy groups, the study provides a clear understanding of their topological features. This research improves our understanding of these groups by offering clear examples and counterexamples, leading to a thorough conclusion about their different topological features. [ABSTRACT FROM AUTHOR]

Let m , n {m,n} be the fixed positive integers and let ℛ {\mathcal{R}} be a ring. In 1978, Herstein proved that a 2-torsion free prime ring ℛ {\mathcal{R}} is commutative if there is a nonzero derivation d of R such that [ d ⁢ ( ϱ ) , d ⁢ ( ξ ) ] = 0 {[d(\varrho),d(\xi)]=0} for all ϱ , ξ ∈ R {\varrho,\xi\in R} . In this article, we study the above mentioned classical result for higher derivations and describe the structure of semiprime rings by using the invariance property of prime ideals under higher derivations. Precisely, apart from proving some other important results, we prove the following. Let ( d i ) i ∈ ℕ {(d_{i})_{i\in\mathbb{N}}} and ( g j ) j ∈ ℕ {(g_{j})_{j\in\mathbb{N}}} be two higher derivations of semiprime ring ℛ {\mathcal{R}} such that [ d n ⁢ ( ϱ ) , g m ⁢ ( ξ ) ] ∈ Z ⁢ ( ℛ ) {[d_{n}(\varrho),g_{m}(\xi)]\in Z(\mathcal{R})} for all ϱ , ξ ∈ ℐ {\varrho,\xi\in\mathcal{I}} , where ℐ {\mathcal{I}} is an ideal of ℛ {\mathcal{R}} . Then either ℛ {\mathcal{R}} is commutative or some linear combination of ( d i ) i ∈ ℕ {(d_{i})_{i\in\mathbb{N}}} sends Z ⁢ ( ℛ ) {Z(\mathcal{R})} to zero or some linear combination of ( g j ) j ∈ ℕ {(g_{j})_{j\in\mathbb{N}}} sends Z ⁢ ( ℛ ) {Z(\mathcal{R})} to zero. We enrich our results with examples that show the necessity of their assumptions. Finally, we conclude our paper with a direction for further research. [ABSTRACT FROM AUTHOR]

Let C be a smooth, projective and geometrically integral curve defined over a finite field F. For each closed point P ∞ of C , let R be the ring of functions that are regular outside P ∞ , and let K be the completion at P ∞ of the function field of C. In order to study groups of the form GL 2 (R) , Serre describes the quotient graph GL 2 (R) ﹨ t , where t is the Bruhat-Tits tree defined from SL 2 (K). In this work we describe the associated quotient graph H ﹨ t for the action on t of the group H = { ( a b c d ) ∈ GL 2 (R) : c ≡ 0 (mod I) } , where I is an ideal of R. These groups play, in the function field context, the same role as the Hecke congruence subgroups of SL 2 (Z). To be precise, we give a explicit formula for the cusp number of H ﹨ t. Then, by using Bass-Serre Theory, we describe H as an amalgamated product of simpler groups, and we also describe its abelianization. [ABSTRACT FROM AUTHOR]

The overall goal of this research is to explore the interactions that exist between the structure of a quotient ring R / P and the behavior of generalized derivations of an arbitrary ring with involution R. To be more exact, we use two generalized derivations of R that satisfy algebraic identities involving the prime ideal P to describe the commutativity of R / P. Besides, we offer instances to demonstrate that the multiple constraints necessary in the assumptions of our theorems are not excessive. [ABSTRACT FROM AUTHOR]

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 . [ABSTRACT FROM AUTHOR]

Professional road cycling requires high endurance and power output for optimal performance. Oval chainrings, a recent innovation, aim to minimize dead spots during pedal revolutions and enhance cycling performance. This study compared the effects of circular (C-ring) and oval (Q-ring) chainrings on power output and maximum oxygen consumption (V̇ O2Max) during an incremental exercise test. Twenty male elite cyclists (mean age ± SD: 19.90 ± 1.86 years) participated in a randomized study. They underwent two incremental maximal tests separated by 48 hours, using both chainring types. The test began with participants pedaling with a 200-watt load in their preferred cadence. During the first sixminute period, the load was increased by 50 watts every two minutes, and then the test continued to increase by 25 watts every two minutes. The test was completed at the point where the participant was exhausted. The results indicated significant differences in favor of Q-rings for V̇ O2Max (p = 0.006) and power output (p = 0.014), while heart rate and cadence remained similar. High correlations were found between V̇ O2Max/power output and heart rate mean/max. Linear regression analysis revealed that power output accounted for 52% (Q-ring) and 64% (C-ring) of the variance in V̇ O2Max. Theoretical advantages of Q-rings include optimizing pedal force distribution, which may enhance mechanical performance. Previous studies have shown mixed results regarding the effects of non-circular chainrings on cycling performance, indicating methodological disparities. Overall, this study suggests that Q-rings may improve road cycling performance by increasing power output. However, further research is needed to explore their efficacy across various cycling scenarios. [ABSTRACT FROM AUTHOR]

Let R be a prime ring of characteristic different from 2 , Q r be its right Martindale quotient ring and C be its extended centroid, α be an automorphism of R , d be a skew derivation of R with associated automorphism α , F and G be two nonzero X -generalized skew derivation of R with associated term (b , α , d) and (b ′ , α , d) , respectively, S be the set of the evaluations of f (x 1 , ... , x n) on R , where f (x 1 , ... , x n) is a non-central multilinear polynomial over C in n non-commuting variables. Let 0 ≠ v ∈ R be such that F (x) x + G (x) x v = 0 for all x ∈ S. Then one of the following statements holds : (a) f (x 1 , ... , x n) 2 is central valued on R and there exist a , a ′ ∈ Q r such that F (x) = a x , G (x) = a ′ x for any x ∈ R with a + a ′ v = 0 ; (b) v ∈ C and F = − v G. In the last part of the paper, we present some applications on the basis of the foregoing proposed result. [ABSTRACT FROM AUTHOR]

Let ℛ be a prime ring, which is not commutative, with involution * and symmetric ring of quotients 풬 s . The aim of the present paper is to describe the structures of a pair of generalized Jordan * -derivations of prime * -rings. Notably, we prove that if a noncommutative prime ring ℛ with involution * admits a couple of generalized Jordan derivations ℱ 1 and ℱ 2 associated with Jordan * -derivations 풟 1 and 풟 2 such that ℱ 1 ⁢ (x) ⁢ x * - x ⁢ ℱ 2 ⁢ (x) = 0 for all x ∈ ℛ , then the following holds: (i) if 풟 1 ⁢ (x) = 풟 2 ⁢ (x) , then ℱ 1 ⁢ (x) = ℱ 2 ⁢ (x) = 0 for all x ∈ ℛ , (ii) if 풟 1 ⁢ (x) ≠ 풟 2 ⁢ (x) , then there exists q ∈ 풬 s such that ℱ 1 ⁢ (x) = x ⁢ q , and ℱ 2 ⁢ (x) = q ⁢ x * for all x ∈ ℛ . Moreover, some related results are also discussed. [ABSTRACT FROM AUTHOR]

Let 풜 be a prime ring equipped with an involution ' * ' of order 2 and let m ≠ n be some fixed positive integers such that 풜 is 2 ⁢ m ⁢ n ⁢ (m + n) ⁢ | m - n | -torsion free. Let 풬 m ⁢ s ⁢ (풜) be the maximal symmetric ring of quotients of 풜 and consider the mappings ℱ and 풢 : 풜 → 풬 m ⁢ s ⁢ (풜) satisfying the relations (m + n) ⁢ ℱ ⁢ (a 2) = 2 ⁢ m ⁢ ℱ ⁢ (a) ⁢ a * + 2 ⁢ n ⁢ a ⁢ ℱ ⁢ (a) and (m + n) ⁢ 풢 ⁢ (a 2) = 2 ⁢ m ⁢ 풢 ⁢ (a) ⁢ a * + 2 ⁢ n ⁢ a ⁢ ℱ ⁢ (a) for all a ∈ 풜 . Using the theory of functional identities and the structure of involutions on matrix algebras, we prove that if ℱ and 풢 are additive, then 풢 = 0 . We also show that, in case ' * ' is any nonidentity anti-automorphism, the same conclusion holds if either ' * ' is not identity on 풵 ⁢ (풜) or 풜 is a PI-ring. [ABSTRACT FROM AUTHOR]

Suppose that R is a prime ring with char(R) ≠ 2 and f(ξ1,... , ξn) 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(ab) = h(a)h(b)+h(a)b+ah(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 ∈ f(R) = {f(ξ1,... , ξn) | ξ1,... , ξn ∈ R}. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring with nonzero identity. In this paper, we introduce and investigate a new class of ideals that is closely related to the class of r -ideals. A proper ideal I of R is said to be a (1 , r) -ideal if whenever nonunit elements a , b , c ∈ R and a b c ∈ I , then a b ∈ I or c ∈ Z (R). Some basic properties of (1 , r) -ideals are studied. For instance, we give a method to construct (1 , r) -ideals that are not r -ideals. Among other things, it is shown that if R admits a (1 , r) -ideal that is not an r -ideal, then R is a local ring. Also, we provide a necessary and sufficient condition in term of (1 , r) -ideals for a ring to be a total quotient ring and we determinate the (1 , r) -ideals of a chained ring. Finally, we give an idea about some (1 , r) -ideals of the localization of rings, the trivial ring extensions and the power series rings to construct nontrivial and original examples of (1 , r) -ideals. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this work, we study the notions of k-ideals and h-ideals of ternary semirings and investigate some of their algebraic properties. Furthermore, we construct a congruence relation with respect to a full k-ideal on a ternary semiring for the purpose of forming a ternary ring from the quotient ternary semiring. [ABSTRACT FROM AUTHOR]

Let H be a complex linear algebraic group with internally graded unipotent radical acting on a complex projective variety X . Given an ample linearisation of the action and an associated Fubini–Study Kähler form which is invariant for a maximal compact subgroup Q of H , we define a notion of moment map for the action of H , and under suitable conditions (that the linearisation is well-adapted and semistability coincides with stability) we describe the (non-reductive) GIT quotient X / / H introduced in (Bérczi et al. in J. Topol. 11(3):826–855, 2018) in terms of this moment map. Using this description we derive formulas for the Betti numbers of X / / H and express the rational cohomology ring of X / / H in terms of the rational cohomology ring of the GIT quotient X / / T H , where T H is a maximal torus in H . We relate intersection pairings on X / / H to intersection pairings on X / / T H , obtaining a residue formula for these pairings on X / / H analogous to the residue formula of (Jeffrey and Kirwan in Topology 34(2):291–327, 1995). As an application, we announce a proof of the Green–Griffiths–Lang and Kobayashi conjectures for projective hypersurfaces with polynomial degree. [ABSTRACT FROM AUTHOR]

Let R be a noncommutative prime ring of characteristic different from 2, Q r be the right Martindale quotient ring of R and C = Z ( Q r) be the extended centroid of R. Suppose that π ( x 1 , ... , x n) is a noncentral multilinear polynomial over C , S = { π (r 1 , ... , r n) | r 1 , ... , r n ∈ R } , F , G and H are three generalized skew-derivations of R associated to the same automorphism α. Let h , f , g be the associated skew derivations respectively of H , F and G , such that h, f, g are commuting with α. We will provide the complete description of H , F and G , in the case F (X) G (X) = H ( X 2) , for all X ∈ S . [ABSTRACT FROM AUTHOR]

Working in a polynomial ring S=k[x1,...,xn]$S={\mathbf {k}}[x_1,\ldots ,x_n]$, where k${\mathbf {k}}$ is an arbitrary commutative ring with 1, we consider the d$d$th Veronese subalgebras R=S(d)$R={S^{(d)}}$, as well as natural R$R$‐submodules M=S(⩾r,d)$M={S^{({\geqslant r},{d})}}$ inside S$S$. We develop and use characteristic‐free theory of Schur functors associated to ribbon skew diagrams as a tool to construct simple GLn(k)$GL_n({\mathbf {k}})$‐equivariant minimal free R$R$‐resolutions for the quotient ring k=R/R+${\mathbf {k}}=R/R_+$ and for these modules M$M$. These also lead to elegant descriptions of ToriR(M,M′)$\operatorname{Tor}^R_i(M,M^{\prime})$ for all i$i$ and HomR(M,M′)$\operatorname{Hom}_R(M,M^{\prime})$ for any pair of these modules M,M′$M,M^{\prime}$. [ABSTRACT FROM AUTHOR]

Let R be a prime ring with center Z(R) and G be a generalized α-derivation of R for α ∈ Aut(R). Let α ∈ R be a nonzero element and n be a fixed positive integer. (i) If aG(x)n ∈ Z(R) for all x ∈ R then aG(x) = 0 for all x ∈ R unless dimc RC = 4. (ii) If aG(x)n ∈ Z(R) for all x ∈ L, where L is a noncommutative Lie ideal of R then aG(x) = 0 for all x ∈ R unless dimc RC = 4. [ABSTRACT FROM AUTHOR]

Let R be a ring, (S,≤) a strictly ordered monoid and ω : S →End(R) a monoid homomorphism. Properties of the ring [[RS,≤, ω]] of skew generalized power series with coef- ficients in R and exponents in S are considered. This paper is devoted to the study of linearly (S, ω)-quasi-Armendariz ring, which is unify the notions of linearly (S, ω)-Armendariz ring and (S, ω)-quasi-Armendariz ring. It is shown that, if R is linearly (S, ω)-quasi-Armendariz ring, U is a nonempty subset in R is a two-sided ideal of R, A = ℝR(U) and ωs|U is surjective for all s ∈ S, then R/A is linearly (S, ω)-quasi-Armendariz. Also, we prove that, R is semiprime if and only if R is reduced linearly (S, ω)-quasi-Armendariz. Moreover, Under a necessary and sufficient conditions, if R is linearly (S, ω)-quasi-Armendariz, then Q is linearly (S, eω)-quasi-Armendariz, where Q is the classical left ring of quotients of R. Consequently, some results of linearly (S, ω)-quasi-Armendariz are given. [ABSTRACT FROM AUTHOR]

The reversible property of rings was introduced by Cohn and has important generalizations in noncommutative ring theory. In this paper, reversibility of rings is investigated in relation with quasinilpotents and idempotents, and our argument is spread out based on this. We call a ring RQnil e-reversible if for any a , b ∈ R , being a ⁢ b = 0 implies b ⁢ a ⁢ e ∈ R qnil for a prescribed idempotent e ∈ R , where R qnil denotes the set of all quasinilpotent elements of R. In the first, we determine the set R qnil for some classes of rings to investigate the structure of Qnil e-reversible rings. In the second, we use R qnil to define Qnil e-reversibility of rings. The notion of Qnil e-reversible ring is a proper generalization of that of e-semicommutative ring, Qnil-semicommutative ring, e-reversible ring and right (left) quasi-duo ring. We obtain some relations between a ring and its quotient rings in terms of Qnil e-reversibility. Applications via some ring extensions and examples illustrating our results are provided. [ABSTRACT FROM AUTHOR]

This paper surveys the theory of functional identities and its applications. No prior knowledge of the theory is required to follow the paper. [ABSTRACT FROM AUTHOR]

We show that any quasi-Gorenstein deformation of a 3-dimensional quasi-Gorenstein Buchsbaum local ring with I -invariant 1 admits a maximal Cohen-Macaulay module, provided it is a quotient of a Gorenstein ring. Such a class of rings includes two instances of unique factorization domains constructed by Marcel-Schenzel and by Imtiaz-Schenzel, respectively. Apart from this result, motivated by the small Cohen-Macaulay conjecture in prime characteristic, we examine a question about when the Frobenius pushforward F ⁎ e (M) of an R -module M comprises a maximal Cohen-Macaulay direct summand in both local and graded cases. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Given a continuous, odd, reducible, and semi-simple |$2$| -dimensional representation |$\bar \rho _{0}$| of |$G_{{\mathbb{Q}},Np}$| over a finite field of odd characteristic |$p$|⁠ , we study the relation between the universal deformation ring of the pseudo-representation corresponding to |$\bar \rho _{0}$| (pseudo-deformation ring) and the big |$p$| -adic Hecke algebra to prove that the maximal reduced quotient of the pseudo-deformation ring is isomorphic to the local component of the big |$p$| -adic Hecke algebra corresponding to |$\bar \rho _{0}$| if a certain global Galois cohomology group has dimension |$1$|⁠. This partially extends the results of Böckle to the case of residually reducible representations. We give an application of our main theorem to the structure of Hecke algebras modulo |$p$|⁠. As another application of our methods and results, we prove a result about non-optimal levels of newforms lifting |$\bar \rho _{0}$| in the spirit of Diamond–Taylor. This also gives a partial answer to a conjecture of Billerey–Menares. [ABSTRACT FROM AUTHOR]

Let k be a positive integer. For a commutative ring R , the ring of dual numbers of k variables over R is the quotient ring R [ x 1 , ... , x k ] / I , where I is the ideal generated by the set { x i x j | i , j ∈ { 1 , ... , k } }. This ring can be viewed as R [ α 1 , ... , α k ] with α i α j = 0 , where α i = x i + I for 1 ≤ i , j ≤ k. We investigate the polynomial functions of R [ α 1 , ... , α k ] whenever R is a finite commutative ring. We derive counting formulas for the number of polynomial functions and polynomial permutations on R [ α 1 , ... , α k ] depending on the order of the pointwise stabilizer of the subring of constants R in the group of polynomial permutations of R [ α 1 , ... , α k ]. Further, we show that the stabilizer group of R is independent of the number of variables k. Moreover, we prove that a function F on R [ α 1 , ... , α k ] is a polynomial function if and only if a system of linear equations on R that depends on F has a solution. [ABSTRACT FROM AUTHOR]

Jacobson investigated the structure of rings with the property that some power of each element is central, in the procedure of the study of commutativity. In this paper, we consider a class of rings in which this property occurs only for central elements, and such rings are called ppnc. We first prove that a noncommutative ppnc ring is infinite, and that a ppnc ring is commutative when it is a K-ring or a locally finite ring. We next study the structure of ppnc rings, and the relation between ppnc rings and related concepts (for example, K-rings, commutative rings and NI rings), through matrix rings, polynomial rings, right quotient rings and direct products. [ABSTRACT FROM AUTHOR]

Let S be a ring. The main objective of this paper is to analyze the structure of quotient rings, which are represented as S/P, where S is an arbitrary ring and P is a prime ideal of S. The paper aims to establish a link between the structure of these rings and the behaviour of traces of symmetric nderivations satisfying some algebraic identities involving prime ideals of an arbitrary ring S. Moreover, as an application of the main result, we investigate the structure of the quotient ring S/P and traces of symmetric n-derivations. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Let R be a ring and M be a monoid with a twisting map f : M × M → U(R) and an action map ω : M → Aut(R). The objective of our work is to extend the reflexive properties of rings by focusing on the crossed product R ∗ M over R. In order to achieve this, we introduce and examine the concept of strongly CM-reflexive. 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 AP P-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. [ABSTRACT FROM AUTHOR]

The primary objective of this paper is to demonstrate that a quotient ring S/G is commutative by studying the behavior of symmetric generalized bi-derivations of S × S that fulfill some algebraic identities concerning a prime ideal. The result of Vukman is extended by proposing identities dealing with different symmetric generalized bi-additive mappings: for all elements l1 of a ring S, the statement λ(l1, l1)l1 + l1λ(l1, l1) - λ(l1, l1) belongs to a prime ideal. [ABSTRACT FROM AUTHOR]

Let S be a ring with a partial action α of a finite group G. We determine when a quotient group of G gives rise to a partial action induced by α on a subring of S. As an application, we show that if S / S α is an α -partial Galois extension and K is a normal subgroup of G , then under certain conditions S α K / S α under the partial action of G / K induced by α , denoted by α G / K , is a partial Galois extension. This result was just shown recently by Bagio et al., applying globalization to derive a partial action of G / K on S α K , totally different from the one presented in this paper arising from a Boolean ring generated by certain idempotents. We further show in either type of induced partial action of a quotient group that under certain conditions S α K / S α is a DeMeyer–Kanzaki α G / K -partial Galois extension whenever S / S α is an α -partial Galois Azumaya extension. A structure theorem for a partial Galois extension is also presented, namely, every partial Galois extension can be decomposed as a direct sum of Galois extensions and possibly a trivial partial Galois extension of type II. [ABSTRACT FROM AUTHOR]