Related papers: Non-associative Ore extensions

The paper concerns perfect diassociative algebras and their implications to the theory of central extensions. It is first established that perfect diassociative algebras have strong ties with universal central extensions. Then, using a…

Let $R$ be an associative unital algebra over a field $k,$ let $p$ be an element of $R,$ and let $R'=R\langle q\mid pqp= p\rangle.$ We obtain normal forms for elements of $R',$ and for elements of $R'$-modules arising by extension of…

In this paper, we introduce the concept of graded extension dimension for a group graded ring R, denoted by gr.ext.dim(R). We prove that when R is strongly graded, its graded extension dimension coincides with the non-graded extension…

In this paper we prove that if R is a left Noetherian and left regular ring such that all finitely generated projective left R-modules are stably free, then the same is true for the completion R[[x;\sigma,\delta]] of any Ore extension…

A prime ring $R$ with extended centroid $C$ is said to be exceptional if both $\text{\rm char}\,R=2$ and $\dim_CRC=4$. Herstein characterized additive subgroups $A$ of a nonexceptional simple ring $R$ satisfying $\big[A, [R,…

For any Lie algebra L over a field, its universal enveloping algebra U(L) can be embedded in a division ring D(L) constructed by Lichtman. If U(L) is an Ore domain, D(L) coincides with its ring of fractions. It is well known that the…

In this work we introduce a new concept, namely, $\tau_{s}$-extending modules (rings) which is torsion-theoretic analogues of extending modules and then we extend many results from extending modules to this new concept. For instance we show…

Let $R$ be a ring, $\sigma:R\to R$ a ring endomorphism, and $\delta:R\to R$ a $\sigma$-derivation. We establish that the Ore extension $R[x;\sigma,\delta]$ satisfies the rank condition if and only if $R$ does. In addition, we prove…

A double Ore extension is a natural generalization of the Ore extension. We prove that a connected graded double Ore extension of an Artin-Schelter regular algebra is Artin-Schelter regular. Some other basic properties such as the…

We revisit the concept of special algebras, also known as \textit{purely inseparable ring extensions}. This concept extends the notion of purely inseparable field extensions to the more general context of extensions of commutative rings. We…

Given an action of a monoid $T$ on a ring $A$ by ring endomorphisms, and an Ore subset $S$ of $T$, a general construction of a fractional skew monoid ring $S^{\rm op} * A * T$ is given, extending the usual constructions of skew group rings…

The polynomial Szemer\'{e}di theorem implies that, for any $\delta \in (0,1)$, any family $\{P_1,\ldots, P_m\} \subset \mathbb{Z}[y]$ of nonconstant polynomials with constant term zero, and any sufficiently large $N$, every subset of…

For a non-commutative domain $R$ and a multiplicatively closed set $S$ the (left) Ore localization of $R$ at $S$ exists if and only if $S$ satisfies the (left) Ore property. Since the concept has been introduced by Ore back in the 1930's,…

Polynomial maps attached to polynomials of an Ore extension are naturally defi ned. In this setting we show the importance of pseudo-linear transformations and give some applications. In particular, factorizations of polynomials in an Ore…

A dynamical analog of the prime ideals for simple non-commutative rings is introduced. We prove a factorization theorem for the dynamical ideals. The result is used to classify the surface knots and links in the smooth 4-dimensional…

Let $R=\mathbf{C}[\xi_1,\xi_2,\ldots]$ be the infinite variable polynomial ring, equipped with the natural action of the infinite symmetric group $\mathfrak{S}$. We classify the $\mathfrak{S}$-primes of $R$, determine the containments among…

Let $R$ be a commutative ring with nonzero identity. Let $\mathcal{I}(R)$ be the set of all ideals of $R$ and let $\delta : \mathcal{I}(R)\longrightarrow \mathcal{I}(R)$ be a function. Then $\delta$ is called an expansion function of ideals…

Consider a commutative monoid $(M,+,0)$ and a biadditive binary operation $\mu \colon M \times M \to M$. We will show that under some additional general assumptions, the operation $\mu$ is automatically both associative and commutative. The…

We present a monomial ideal $\mathfrak{a} \subset S$ such that $S/\mathfrak{a}$ is not Golod, even though the product on its Koszul homology is trivial. This constitutes a counterexample to a well-known result by Berglund and J\"ollenbeck…

In this paper, we introduce a new graph whose vertices are the nonzero zero-divisors of commutative ring $R$ and for distincts elements $x$ and $y$ in the set $Z(R)^{\star}$ of the nonzero zero-divisors of $R$, $x$ and $y$ are adjacent if…