Artinian and noetherian partial skew groupoid rings

Simplicity and Chain Conditions for Ultragraph Leavitt Path Algebras via Partial Skew Group Ring Theory

“…In [14], chain conditions were described for partial skew groupoid rings. As an application, a new proof of the criterion for a Leavitt path algebra to be artinian is given.…”

“…The theory of partial skew group rings has been in constant development recently; see, for example, [5,7], where simplicity criteria are described, [14], where chain conditions are studied, and [4] (and the 283 references therein cited), where most of the recent developments in the theory are compiled. In our case we use partial skew ring theory to characterize artinian ultragraph path algebras and give simplicity criteria for these algebras.…”

Recent developments around partial actions

“…In a recent preprint [247] a rather general situation of a unital partial (in particular, global) action of a groupoid G on a non-necessarily associative ring R was considered. In particular, it was proved that if the partial action is unital and R is alternative, then R * G is left (or right) artinian if and only if so too is R and R g = {0} for all but finitely many g ∈ mor(G).…”

Isomorphism Theorems for Groupoids and Some Applications

“…Recently, some applications of groupoids to the study of partial actions are presented in different branches, for instance, in [6] the author constructs a Birget-Rhodes expansion G BR associated with an ordered groupoid G and shows that it classifies partial actions of G on sets, in the topological context in [7] is treated the globalization problem, connections between partial actions of groups and groupoids are given in [8,9]. Also, ring theoretic and cohomological results of global and partial actions of groupoids on algebras are obtained in [10][11][12][13][14][15][16]. Galois theoretic results for groupoid actions are obtained in [12,[17][18][19].…”