arXiv subjects
Publications and source records attributed to Patrik Lundström.
We prove an analogue of Hilbert's basis theorem for Poisson Ore extensions and Poisson Laurent Ore extensions. We also obtain corresponding results for iterated Poisson Ore extensions and iterated Poisson Laurent Ore extensions associated to commuting Poisson-pairs. Finally, we give examples of Poisson Ore extensions that are Poisson-Noetherian without being Noetherian as ordinary algebras.
Let $R$ be a nonzero associative unital ring, let $G$ be a group, and let $ρ$ be a preorder on $\{1,\ldots,n\}$. A $G$-grading on $ρ$ induces a very good $G$-grading on the structural matrix ring $M_n(ρ,R)$. We show that, for each of the properties trivial, symmetric, epsilon-strong and strong, the grading on $ρ$ has the property if and only if the induced ring grading does. The epsilon-crossed product and crossed product properties pass from $ρ$ to the ring, but the converses fail in general. We also give a concrete criterion for epsilon-strongness and show that a very good $G$-grading on $M_n(ρ,R)$ that is strong satisfies $|G|\leq n$. When $ρ$ is an equivalence relation and the neutral component is diagonal, very good gradings correspond bijectively to free partial actions of $G$ on $\{1,\ldots,n\}$ with orbit relation $ρ$. These gradings are epsilon-crossed products, and over a field the correspondence gives a classification up to graded algebra isomorphism.
We develop a theory of separable ring extensions and separable functors for nonunital rings in the setting of firm modules. We prove nonunital analogues of classical results on functorial separability and semisimplicity, and apply these results to obtain a locally unital version of Maschke's theorem for group rings.
We prove that if $R$ is a ring that is object unital and strongly graded by a groupoid $Γ$, and if $Δ$ is a wide subgroupoid of $Γ$, then $R/R_Δ$ is separable if and only if, for each $e \in Γ_0$, there exist $f \in [e]$ and $r \in C_{R_0}(R_Λ) := \{ x \in R_0 \mid xy = yx \text{ for all } y \in R_Λ\}$ with ${\rm tr}_{Γ/Δ}^f(r) = 1_{R_f}$. Here, $Γ_0$ denotes the set of objects of $Γ$, $[e]$ the connected component of $Γ_0$ containing $e$, $Λ$ the isotropy groupoid of $Δ$, and ${\rm tr}_{Γ/Δ}^f$ the relative trace map at $f$. This result simultaneously generalizes earlier theorems on separability for matrix rings and group-graded rings due to DeMeyer-Ingraham, N{\v a}st{\v a}sescu, Van den Bergh, Van Oystaeyen, Miyashita, Theohari-Apostolidi, and Vavatsoulas, as well as results on groupoid-graded rings due to Cala, Lundström, and Pinedo. As an application, we consider separability for object crossed products, including object twisted groupoid rings, classical groupoid rings and matrix rings, as well as crossed product algebras defined by infinite separable field extensions.
Let $G$ be a group with identity element $e$, and suppose that $S$ is an associative $G$-graded ring that is not necessarily unital. In the case where $G$ is an ordered group, we show that a graded ideal is prime if and only if it is graded prime. Consequently, in that setting, a graded ring is prime if and only if it is graded prime. For any group $G$, if $S$ is what we call ideally symmetrically $G$-graded, then we show that there is a bijective correspondence between the $G$-graded prime ideals of $S$ and the $G$-prime ideals of $S_e$. We use this correspondence in the case where $G$ is ordered and $S$ is ideally symmetrically $G$-graded to show that $S$ is prime if and only if $S_e$ is $G$-prime. These results generalize classical theorems by Năstăsescu and Van Oystaeyen to a nonunital setting. As applications, we provide a new proof of a primeness criterion for Leavitt path rings and establish conditions for primeness of symmetrically $G$-graded subrings of group rings over fully idempotent rings.
Given a set $A$ and an abelian group $B$ with operators in $A$, in the sense of Krull and Noether, we introduce the Ore group extension $B[x; σ_B, δ_B]$ as the additive group $B[x]$, with $A[x]$ as a set of operators. Here, the action of $A[x]$ on $B[x]$ is defined by mimicking the multiplication used in the classical case where $A$ and $B$ are the same ring. We derive generalizations of Vandermonde's and Leibniz's identities for this construction, and they are then used to establish associativity criteria. Additionally, we prove a version of Hilbert's basis theorem for this structure, under the assumption that the action of $A$ on $B$ is what we call weakly $s$-unital. Finally, we apply these results to the case where $B$ is a left module over a ring $A$, and specifically to the case where $A$ and $B$ coincide with a non-associative ring which is left distributive but not necessarily right distributive.
Let $X$ be a magma, that is a set equipped with a binary operation, and consider a function $α: X \to X$. We that $X$ is Hom-associative if for all $x,y,z \in X$, the equality $α(x)(yz) = (xy) α(z)$ holds. For every isomorphism class of magmas of order two, we determine all functions $α$ making $X$ Hom-associative. Furthermore, we find all such $α$ that are endomorphisms of $X$. We also consider versions of these results where the binary operation on $X$ as well as the function $α$ may be only partially defined. We use our findings to construct examples of Hom-associative and multiplicative magma algebras.
We investigate properties of group gradings on matrix rings $M_n(R)$, where $R$ is an associative unital ring and $n$ is a positive integer. More precisely, we introduce very good gradings and show that any very good grading on $M_n(R)$ is necessarily epsilon-strong. We also identify a condition that is sufficient to guarantee that $M_n(R)$ is an epsilon-crossed product, i.e. isomorphic to a crossed product associated with a unital twisted partial action. In the case where $R$ has IBN, we are able to provide a characterization of when $M_n(R)$ is an epsilon-crossed product. Our results are illustrated by several examples.
Suppose that $R$ is an associative unital ring and that $E=(E^0,E^1,r,s)$ is a directed graph. Utilizing results from graded ring theory we show, that the associated Leavitt path algebra $L_R(E)$ is simple if and only if $R$ is simple, $E^0$ has no nontrivial hereditary and saturated subset, and every cycle in $E$ has an exit. We also give a complete description of the center of a simple Leavitt path algebra.
In this article, we study Ore extensions of non-unital associative rings. We provide a characterization of simple non-unital differential polynomial rings $R[x;δ]$, under the hypothesis that $R$ is $s$-unital and $\ker(δ)$ contains a nonzero idempotent. This result generalizes a result by Öinert, Richter and Silvestrov from the unital setting. We also present a family of examples of simple non-unital differential polynomial rings.
We provide a short proof, not utilizing complex numbers, for the solution set of homogeneous second order linear differential equations with constant coefficients.
We present an elementary self-contained folkloristic proof, using limits of primitives of Bernstein polynomials, for the existence of primitive functions of continuous functions defined on the unit interval.
We obtain criteria for when a ring with enough idempotents is left/right artinian or noetherian in terms of local criteria defined by the associated complete set of idempotents for the ring. We apply these criteria to object unital category graded rings in general and, in particular, to the class of skew category algebras. Thereby, we generalize results by Nastasescu-van Oystaeyen, Bell, Park and Zelmanov from the group graded case to groupoid, and in some cases category, gradings.
We present a streamlined, slightly modified version, in the two-variable situation, of a beautiful, but not so well known, theory by Bögel, already from the 1930s, on an alternative higher dimensional calculus of real functions, a double calculus, which includes two-variable extensions of many classical results from single variable calculus, such as Rolle's theorem, Lagrange's mean value theorem, Cauchy's mean value theorem, Fermat's extremum theorem, the first derivative test, and the first and second fundamental theorems of calculus.
In a previous article (see \cite{CNP}), we introduced and analyzed ring-theoretic properties of object unital $\mathcal{G}$-graded rings $R$, where $\mathcal{G}$ is a groupoid. In the present article, we analyze the category $\grmod$ of unitary $\G$-graded modules over such rings. Following ideas developed earlier by one of the authors in \cite{lundstrom2004}, we analyze the forgetful functor $U \colon \grmod \to \rmod$ and aim to determine properties $\mathcal{P}$ for which the following implications are valid for modules $M$ in $\grmod$: $M$ is $\mathcal{P}$ $\Rightarrow$ $U(M)$ is $\mathcal{P}$; $U(M)$ is $\mathcal{P}$ $\Rightarrow$ $M$ is $\mathcal{P}$. Here we treat the cases when $\mathcal{P}$ is any of the properties: direct summand, projective, injective, free, simple and semisimple. Moreover, graded versions of results concerning classical module theory are established, as well as some structural properties related to the category $\grmod$.
In this article we generalize a classical result by Passman on primeness of unital strongly group graded rings to the class of nearly epsilon-strongly group graded rings which are not necessarily unital. Using this result, we obtain (i) a characterization of prime $s$-unital strongly group graded rings, and, in particular, of infinite matrix rings and of group rings over $s$-unital rings, thereby generalizing a well-known result by Connell; (ii) characterizations of prime $s$-unital partial skew group rings and of prime unital partial crossed products; (iii) a generalization of the well-known characterizations of prime Leavitt path algebras, by Larki and by Abrams-Bell-Rangaswamy.
Let $R$ be a unital ring, let $E$ be a directed graph and recall that the Leavitt path algebra $L_R(E)$ carries a natural $\mathbb{Z}$-gradation. We show that $L_R(E)$ is strongly $\mathbb{Z}$-graded if and only if $E$ is row-finite, has no sink, and satisfies Condition (Y). Our result generalizes a recent result by Clark, Hazrat and Rigby, and the proof is short and self-contained.
In this article, we continue our study of category dynamical systems, that is functors $s$ from a category $G$ to $\Top^{\op}$, and their corresponding skew category algebras. Suppose that the spaces $s(e)$, for $e \in \ob(G)$, are compact Hausdorff. We show that if (i) the skew category algebra is simple, then (ii) $G$ is inverse connected, (iii) $s$ is minimal and (iv) $s$ is faithful. We also show that if $G$ is a locally abelian groupoid, then (i) is equivalent to (ii), (iii) and (iv). Thereby, we generalize results by Öinert for skew group algebras to a large class of skew category algebras.