Narrow Results

In this paper, we investigate the property of schematicness introduced by Van Oystaeyen and Willaert [F. Van Oystaeyen and L. Willaert, Grothendieck topology, coherent sheaves and Serre’s theorem for schematic algebras, J. Pure Appl. Algebra104(1–3) (1995) 109–122] in the setting of skew Ore polynomials of higher order generated by homogenous quadratic relations defined by Golovashkin and Maksimov [A. V. Golovashkin and V. M. Maksimov, Skew Ore polynomials of higher orders generated by homogeneous quadratic relations, Russian Math. Surveys53(2) (1998) 384–386]. We establish sufficient conditions to guarantee whether some of these algebras are schematic or not.

In this paper, the simple modules over the second quantized Weyl algebras at roots of unity over an algebraically closed field are classified.

In this paper, the Azumaya loci of the coordinate ring ${{𝒪}}_q(𝔬{𝕂}^{2n})$ of the quantum Euclidean $2n$-space are computed assuming the parameter $q$ is an odd $m$th root of unity.

In this paper, we are interested in the class of $(\lambda ,{\delta }_{\sigma })$-constacyclic codes with derivation over the ring ${ℛ}={\mathbb{Z}}_4+u{\mathbb{Z}}_4$, where $u^2=0$, $\sigma$ is an automorphism of $R$ and ${\delta }_{\sigma }$ is a $\sigma$-derivation (skew derivation). Similar to the case of cyclic codes and their generalizations, we proved a one-to-one correspondence between $(\lambda ,{\delta }_{\sigma })$-constacyclic codes and the left ${ℛ}[x,{\delta }_{\sigma }]$-submodules of ${{ℛ}}_{\lambda }:={ℛ}[x,{\delta }_{\sigma }]/〈x^n-\lambda 〉.$ As an application, we derived many characterizations of these types of codes. Finally, we provided some examples as applications of the theory developed here.

In the author's work [S. A. Annin, Attached primes over noncommutative rings, J. Pure Appl. Algebra212 (2008) 510–521], a theory of attached prime ideals in noncommutative rings was developed as a natural generalization of the classical notions of attached primes and secondary representations that were first introduced in 1973 as a dual theory to the associated primes and primary decomposition in commutative algebra (see [I. G. Macdonald, Secondary representation of modules over a commutative ring, Sympos. Math.11 (1973) 23–43]). Associated primes over noncommutative rings have been thoroughly studied and developed for a variety of applications, including skew polynomial rings: see [S. A. Annin, Associated primes over skew polynomial rings, Commun. Algebra30(5) (2002) 2511–2528; and S. A. Annin, Associated primes over Ore extension rings, J. Algebra Appl.3(2) (2004) 193–205]. Motivated by this background, the present article addresses the behavior of the attached prime ideals of inverse polynomial modules over skew polynomial rings. The goal is to determine the attached primes of an inverse polynomial module M[x^-1] over a skew polynomial ring R[x;σ] in terms of the attached primes of the base module M_R. This study was completed in the commutative setting for the class of representable modules in [L. Melkersson, Content and inverse polynomials on artinian modules, Commun. Algebra26(4) (1998) 1141–1145], and the generalization to noncommutative rings turns out to be quite non-trivial in that one must either work with a Bass module M_R or a right perfect ring R in order to achieve the desired statement even when no twist is present in the polynomial ring "Let M_R be a module over any ring R. If M[x^-1]_R is a completely σ-compatible Bass module, then Att(M[x^-1]_S) = {𝔭[x] : 𝔭 ∈ Att(M_R)}." The sharpness of the results are illustrated through the use of several illuminating examples.

Let R be an associative ring with two-sided multiplicative identity. In this paper, in the case that R is a commutative α-compatible ring, we compare the diameter (and girth) of the zero-divisor graphs Γ(R) and Γ(R[x;α, δ]). Moreover, we study the zero-divisors of the Öre extension ring R[x;α, δ], whenever R is reversible and (α, δ)-compatible.

Let R be a ring with an endomorphism α and α-derivation δ. By [A. R. Nasr-Isfahani and A. Moussavi, Ore extensions of skew Armendariz rings, Comm. Algebra 36(2) (2008) 508–522], a ring R is called a skew Armendariz ring, if for polynomials f(x) = a₀ + a₁ x + ⋯ + a_nxⁿ, g(x) = b₀+b₁x + ⋯ + b_mx^m in R[x; α, δ], f(x)g(x) = 0 implies a₀b_j = 0 for each 0 ≤ j ≤ m. In this paper, radicals of the skew polynomial ring R[x; α, δ], in terms of a skew Armendariz ring R, is determined. We prove that several properties transfer between R and R[x; α, δ], in case R is an α-compatible skew Armendariz ring. We also identify some "relatively maximal" skew Armendariz subrings of matrix rings, and obtain a necessary and sufficient condition for a trivial extension to be skew Armendariz. Consequently, new families of non-reduced skew Armendariz rings are presented and several known results related to Armendariz rings and skew polynomial rings will be extended and unified.

Antoine studied the structure of the set of nilpotent elements in Armendariz rings and introduced the concept of nil-Armendariz property as a generalization. Hong et al. studied Armendariz property on skew polynomial rings and introduced the notion of an α-Armendariz ring, where α is a ring monomorphism. In this paper, we investigate the structure of the set of α-nilpotent elements in α-Armendariz rings and introduce an α-nil-Armendariz ring. We examine the set of -nilpotent elements in a skew polynomial ring R[x;α], where is the monomorphism induced by the monomorphism α of an α-Armendariz ring R. We prove that every polynomial with α-nilpotent coefficients in a ring R is -nilpotent when R is of bounded index of α-nilpotency, and moreover, R is shown to be α-nil-Armendariz in this situation. We also characterize the structure of the set of α-nilpotent elements in α-nil-Armendariz rings, and investigate the relations between α-(nil-)Armendariz property and other standard ring theoretic properties.

In this paper, we continue to study zero-divisor properties of skew polynomial rings $R[x;\alpha ,\delta ]$, where $R$ is an associative ring equipped with an endomorphism $\alpha$ and an $\alpha$-derivation $\delta$. For an associative ring $R$, the undirected zero-divisor graph of $R$ is the graph $\mathrm{\Gamma }(R)$ such that the vertices of $\mathrm{\Gamma }(R)$ are all the nonzero zero-divisors of $R$ and two distinct vertices $x$ and $y$ are connected by an edge if and only if $xy=0$ or $yx=0$. As an application of reversible rings, we investigate the interplay between the ring-theoretical properties of a skew polynomial ring $R[x;\alpha ,\delta ]$ and the graph-theoretical properties of its zero-divisor graph $\mathrm{\Gamma }(R[x;\alpha ,\delta ])$. Our goal in this paper is to give a characterization of the possible diameters of $(\mathrm{\Gamma }(R[x;\alpha ,\delta ]))$ in terms of the diameter of $\mathrm{\Gamma }(R)$, when the base ring $R$ is reversible and also have the $(\alpha ,\delta )$-compatible property. We also completely describe the associative rings all of whose zero-divisor graphs of skew polynomials are complete.

In this paper, we extend work from [M. Vancliff and P. P. Veerapen, Generalizing the notion of rank to noncommutative quadratic forms, in Noncommutative Birational Geometry, Representations and Combinatorics, eds. A. Berenstein and V. Retakh, Contemporary Mathematics, Vol. 592 (2013), pp. 241–250], where a notion of rank, called $\mu$-rank, was proposed for noncommutative quadratic forms on two and three generators. In particular, we provide a definition of $\mu$-rank one and two for noncommutative quadratic forms on four generators. We apply this definition to determine the number of point modules over certain quadratic AS-regular algebras of global dimension four.

We study the structure of the set of nilpotent elements in extended semicommutative rings and introduce nil α-semicommutative rings as a generalization. We resolve the structure of nil α-semicommutative rings and obtain various necessary or sufficient conditions for a ring to be nil α-semicommutative, unifying and generalizing a number of known commutative-like conditions in special cases. We also classify which of the standard nilpotence properties on polynomial rings pass to skew polynomial ring. Constructing various examples, we classify how the nil α-semicommutative rings behaves under various ring extensions. Also, we consider the nil-Armendariz condition on a skew polynomial ring.

Let $R$ be an associative ring with identity, $S$ a monoid and $\omega :S\to \mathrm{\text{End}}(R)$ a monoid homomorphism. When $S$ is a u.p.-monoid and $R$ is a reversible $S$-compatible ring, then we observe that $R$ satisfies a McCoy-type property, in the context of skew monoid ring $R\ast S$. We introduce and study the $(S,\omega )$-McCoy condition on $R$, a generalization of the standard McCoy condition from polynomial rings to skew monoid rings. Several examples of reversible $S$-compatible rings and also various examples of $(S,\omega )$-McCoy rings are provided. As an application of $(S,\omega )$-McCoy rings, we investigate the interplay between the ring-theoretical properties of a general skew monoid ring $R\ast S$ and the graph-theoretical properties of its zero-divisor graph $\overline{\mathrm{\Gamma }}(R\ast S)$.

In this paper, the skew constacyclic codes over finite non-chain ring ${ℛ}={𝔽}_q+u{𝔽}_q+v{𝔽}_q$, where $q=p^m$, $p$ is an odd prime and $u^2=u,v^2=v,uv=vu=0$, are studied. We show that the Gray image of a skew $\alpha$-constacyclic code of length $n$ over ${ℛ}$ is a skew quasi-twisted code of length $3n$ over ${𝔽}_q$ of index 3. Further, the structural properties of skew constacyclic codes over ${ℛ}$ are obtained by the decomposition method.

In this paper, we study skew cyclic codes over the ring ${ℛ}={𝔽}_q[u,v,w]/〈u^2-1,v^2-1,w^2-1,uv-vu,vw-wv,wu-uw〉$ where $q=p^m$; $p$ is an odd prime. Here, the structural properties, generating polynomials and idempotent generators of the skew cyclic codes are obtained. Also, we show that the Gray images of skew cyclic codes of length $n$ over ${ℛ}$ are skew 8-quasi-cyclic codes of length $8n$.