Ring theoretic properties of partial crossed products and related themes

Abstract

In this paper we work with unital twisted partial actions. We investigate ring theoretic properties of partial crossed products as artinianity, noetherianity, perfect property, semilocalproperty, semiprimary property and we also study the Krull dimension. Moreover, we consider triangular matrix representation of partial skew group rings, weak and global dimensions of partial crossed products Also we study when the partial crossed products are Frobenius and symmetric algebras.

Recent developments around partial actions

“…Thus facts about globalization from [4] were used in [237] with respect to K -theory of reduced C * -algebras of 0-F-inverse semigroups, in [225] in the K -theoretic study of reduced crossed products attached to totally disconnected dynamical systems, and in [26] for partial flows with application to Lyapunov functions. In addition, globalizable partial actions were essential for the development of Galois Theory of partial group actions in [123], for the elaboration of the concept of a partial Hopf (co)action in [72], as well as in a series of ring theoretic and Galois theoretic investigations in [27,29,30,32,35,39,41,49,50,69,77,98,99,101,104,106,171,176,252,253].…”

“…This algebraic concept was applied to Hecke algebras in [153], where, among other results, it was proved that given a field κ of characteristic 0, a group G and subgroups H, N ⊆ G with N normal in G and H normal in N , there is a twisted partial action θ of G/N on the group algebra κ(N /H ) such that the Hecke algebra H(G, H ) is isomorphic to the crossed product κ(N /H ) * θ G/N . The globalization problem for twisted partial group actions was investigated in [122], whereas other algebraic results on twisted partial actions on rings and corresponding crossed products were obtained in [39,44,49,50,253].…”

“…More recently, in [49] partial crossed product by twisted partial actions were considered with respect to the artinian, noetherian, semilocal, perfect and semiprimary properties, under some assumptions on the twisted partial actions, such as to be of finite type. If the twisted partial action is globalizable, then the above ring theoretic properties for the partial crossed products were related to those for the corresponding global crossed product.…”

“…If the twisted partial action is globalizable, then the above ring theoretic properties for the partial crossed products were related to those for the corresponding global crossed product. Moreover, the Krull dimension of partial crossed products was also studied in [49], as well as the global dimension and the weak global dimension. Furthermore, triangular matrix representations of partial skew group rings were also considered, and, in addition, it was shown with certain assumptions that the ring under the twisted partial action is Frobenius or symmetric, then so too is the crossed product.…”

Ring theoretic properties of partial skew groupoid rings with applications to Leavitt path algebras

“…The properties of B that are studied are the following: noetherianity, von Neumann regularity, the Jacobson radical, perfect, semiprimary and Krull dimension. One of the strategies used to prove the results is to apply the ring theoretic properties proved in [11] to B since it is a skew partial group ring.…”