Let $D$ be a division algebra, finite-dimensional over its center, and $R=D[t;\;\sigma,\delta ]$ a skew polynomial ring.

Using skew polynomials $f\in R$, we construct division algebras and maximum rank distance codes consisting of matrices with entries in a noncommutative division algebra or field. These include Jha Johnson semifields, and the classes of classical and twisted Gabidulin codes constructed by Sheekey.

Let D be a division ring and N be a subnormal subgroup of the multiplicative group $D^*$. We show that if N contains a nonabelian solvable subgroup, then N contains a nonabelian free subgroup.

We study the discrete dynamics of standard (or left) polynomials $f(x)$ over division rings D. We define their fixed points to be the points $\lambda \in D$ for which $f^{\circ n}(\lambda )=\lambda $ for any $n \in \mathbb {N}$ , where $f^{\circ n}(x)$ is defined recursively by $f^{\circ n}(x)=f(f^{\circ (n-1)}(x))$ and $f^{\circ 1}(x)=f(x)$ . Periodic points are similarly defined. We prove that $\lambda $ is a fixed point of $f(x)$ if and only if $f(\lambda )=\lambda $ , which enables the use of known results from the theory of polynomial equations, to conclude that any polynomial of degree $m \geq 2$ has at most m conjugacy classes of fixed points. We also show that in general, periodic points do not behave as in the commutative case. We provide a sufficient condition for periodic points to behave as expected.

Given an action ${\varphi }$ of inverse semigroup S on a ring A (with domain of ${\varphi }(s)$ denoted by $D_{s^*}$ ), we show that if the ideals $D_e$ , with e an idempotent, are unital, then the skew inverse semigroup ring $A\rtimes S$ can be realized as the convolution algebra of an ample groupoid with coefficients in a sheaf of (unital) rings. Conversely, we show that the convolution algebra of an ample groupoid with coefficients in a sheaf of rings is isomorphic to a skew inverse semigroup ring of this sort. We recover known results in the literature for Steinberg algebras over a field as special cases.

Let D be a unital associative division ring and D[t, σ, δ] be a skew polynomial ring, where σ is an endomorphism of D and δ a left σ-derivation. For each f ϵ D[t, σ, δ] of degree m > 1 with a unit as leading coefficient, there exists a unital nonassociative algebra whose behaviour reflects the properties of f. These algebras yield canonical examples of right division algebras when f is irreducible. The structure of their right nucleus depends on the choice of f. In the classical literature, this nucleus appears as the eigenspace of f and is used to investigate the irreducible factors of f. We give necessary and sufficient criteria for skew polynomials of low degree to be irreducible. These yield examples of new division algebras Sf.

Nakayama automorphisms play an important role in the fields of noncommutative algebraic geometry and noncommutative invariant theory. However, their computations are not easy in general. We compute the Nakayama automorphism ν of an Ore extension R[x; σ, δ] over a polynomial algebra R in n variables for an arbitrary n. The formula of ν is obtained explicitly. When σ is not the identity map, the invariant EG is also investigated in terms of Zhang’s twist, where G is a cyclic group sharing the same order with σ.

We study an operation, that we call lifting, creating nonisomorphic monomial curves from a single monomial curve. Our main result says that all but finitely many liftings of a monomial curve have Cohen–Macaulay tangent cones even if the tangent cone of the original curve is not Cohen–Macaulay. This implies that the Betti sequence of the tangent cone is eventually constant under this operation. Moreover, all liftings have Cohen–Macaulay tangent cones when the original monomial curve has a Cohen–Macaulay tangent cone. In this case, all the Betti sequences are just the Betti sequence of the original curve.

Let $R$ be an associative ring with identity. First we prove some results about zero-divisor graphs of reversible rings. Then we study the zero-divisors of the skew power series ring $R\left[\!\left[ x;\,\alpha \right]\!\right]$, whenever $R$ is reversible $\alpha$-compatible. Moreover, we compare the diameter and girth of the zero-divisor graphs of $\Gamma \left( R \right),\,\Gamma \left( R[x;\,\alpha ,\,\delta ] \right)$, and $\Gamma \left( R\left[\!\left[ x;\,\alpha \right]\!\right] \right)$, when $R$ is reversible and $\left( \alpha ,\,\delta \right)$-compatible.

We study an associative algebra $A$ over an arbitrary field that is a sum of two subalgebras $B$ and $C$ (i.e., $A\,=\,B+C$ ). We show that if $B$ is a right or left Artinian $PI$ algebra and $C$ is a $PI$ algebra, then $A$ is a $PI$ algebra. Additionally, we generalize this result for semiprime algebras $A$ . Consider the class of all semisimple finite dimensional algebras $A\,=\,B+C$ for some subalgebras $B$ and $C$ that satisfy given polynomial identities $f\,=\,0$ and $g\,=\,0$ , respectively. We prove that all algebras in this class satisfy a common polynomial identity.

We show that every finitely generated algebra that is a finitely generated module over a finitely generated commutative subalgebra is an automaton algebra in the sense of Ufnarovskii.

We consider valued fields with a distinguished contractive map as valued modules over the Ore ring of difference operators. We prove quantifier elimination for separably closed valued fields with the Frobenius map, in the pure module language augmented with functions yielding components for a p-basis and a chain of subgroups indexed by the valuation group.

The trace on matrix rings, along with the augmentation map and Kaplansky trace on group rings, are some of the many examples of linear functions on algebras that vanish on all commutators. We generalize and unify these examples by studying traces on (contracted) semigroup rings over commutative rings. We show that every such ring admits a minimal trace (i.e., one that vanishes only on sums of commutators), classify all minimal traces on these rings, and give applications to various classes of semigroup rings and quotients thereof. We then study traces on Leavitt path algebras (which are quotients of contracted semigroup rings), where we describe all linear traces in terms of central maps on graph inverse semigroups and, under mild assumptions, those Leavitt path algebras that admit faithful traces.

Let R be a ring and M a monoid with twisting f:M × M → U(R) and action ω: M→ Aut(R). We introduce and study the concepts of CM-Armendariz and CM-quasi-Armendariz rings to generalise various Armendariz and quasi-Armendariz properties of rings by working on the context of the crossed product R*M over R. The following results are proved: (1) If M is a u.p.-monoid, then any M-rigid ring R is CM-Armendariz; (2) if I is a reduced ideal of an M-compatible ring R with M a strictly totally ordered monoid, then R/I being CM-Armendariz implies that R is CM-Armendariz; (3) if M is a u.p.-monoid and R is a semiprime ring, then R is CM-quasi-Armendariz. These results generalise and unify many known results on this subject.

We provide necessary and sufficient conditions for a skew polynomial ring of derivation type to be semiprimitive when the base ring has no nonzero nil ideals. This extends existing results on the Jacobson radical of skew polynomial rings of derivation type.

We consider finite groups acting on quantum (or skew) polynomial rings. Deformations of the semidirect product of the quantum polynomial ring with the acting group extend symplectic reflection algebras and graded Hecke algebras to the quantum setting over a field of arbitrary characteristic. We give necessary and sufficient conditions for such algebras to satisfy a Poincaré–Birkhoff–Witt property using the theory of noncommutative Gröbner bases. We include applications to the case of abelian groups and the case of groups acting on coordinate rings of quantum planes. In addition, we classify graded automorphisms of the coordinate ring of quantum 3-space. In characteristic zero, Hochschild cohomology gives an elegant description of the Poincaré–Birkhoff–Witt conditions.

Non-degenerate monoids of skew type are considered. This is a class of monoids S defined by n generators and quadratic relations of certain type, which includes the class of monoids yielding set-theoretic solutions of the quantum Yang–Baxter equation, also called binomial monoids (or monoids of I-type with square-free defining relations). It is shown that under any degree-lexicographic order on the associated free monoid FMn. of rank n the set of normal forms of elements of S is a regular language in FMn. As one of the key ingredients of the proof, it is shown that an identity of the form xN yN = yN xN holds in S. The latter is derived via an investigation of the structure of S viewed as a semigroup of matrices over a field. It also follows that the semigroup algebra K[S] is a finite module over a finitely generated commutative subalgebra of the form K[A] for a submonoid A of S.

A type of generalized higher derivation consisting of a collection of self-mappings of a ring associated with a monoid, and here called a D-structure, is studied. Such structures were previously used to define various kinds of ‘skew’ or ‘twisted’ monoid rings. We show how certain gradings by monoids define D-structures. The monoid ring defined by such a structure corresponding to a group-grading is the variant of the group ring introduced by Năstăsescu, while in the case of a cyclic group of order two, the form of the D-structure itself yields some gradability criteria of Bakhturin and Parmenter. A partial description is obtained of the D-structures associated with infinite cyclic monoids.

Let R be a ring, S a strictly ordered monoid, and ω:S→End(R) a monoid homomorphism. The skew generalized power series ring R[[S,ω]] is a common generalization of (skew) polynomial rings, (skew) power series rings, (skew) Laurent polynomial rings, (skew) group rings, and Mal’cev–Neumann Laurent series rings. We study the (S,ω)-Armendariz condition on R, a generalization of the standard Armendariz condition from polynomials to skew generalized power series. We resolve the structure of (S,ω)-Armendariz rings and obtain various necessary or sufficient conditions for a ring to be (S,ω)-Armendariz, unifying and generalizing a number of known Armendariz-like conditions in the aforementioned special cases. As particular cases of our general results we obtain several new theorems on the Armendariz condition; for example, left uniserial rings are Armendariz. We also characterize when a skew generalized power series ring is reduced or semicommutative, and we obtain partial characterizations for it to be reversible or 2-primal.

A ring R is called a left APP-ring if the left annihilator lR(Ra) is pure as a left ideal of R for every a ∈ R; R is called (left principally) quasi-Baer if the left annihilator of every (principal) left ideal of R is generated by an idempotent. Let R be a ring and M an ordered monoid. Assume that there is a monoid homomorphism φ: M ⟶ Aut(R). We give a necessary and sufficient condition for the skew monoid ring (induced by φ) to be left APP (left principally quasi-Baer, quasi-Baer, respectively).