Let denote the twisted smash product of an arbitrary algebra A and a Hopf algebra H over a field. We present an analogue of the celebrated Blattner-Montgomery duality theorem for , and as an application we establish the relationship between the homological dimensions of and A if H and its dual H* are both semisimple.
Let 𝓢 be a class of finitely presented R-modules such that R∈ 𝓢 and 𝓢 has a subset 𝓢* with the property that for any U∈ 𝓢 there is a U*∈ 𝓢* with U* ≅ U. We show that the class of 𝓢-pure injective R-modules is preenveloping. As an application, we deduce that the left global 𝓢-pure projective dimension of R is equal to its left global 𝓢-pure injective dimension. As our main result, we prove that, in fact, the class of 𝓢-pure injective R-modules is enveloping.
We introduce the concept of relative Hom-Hopf modules and investigate their structure in a monoidal category . More particularly, the fundamental theorem for relative Hom-Hopf modules is proved under the assumption that the Hom-comodule algebra is cleft. Moreover, Maschke’s theorem for relative Hom-Hopf modules is established when there is a multiplicative total Hom-integral.
Let be a finite abelian group with identity element and be an infinite dimensional -homogeneous vector space over a field of characteristic . Let be the Grassmann algebra generated by . It follows that is a -graded algebra. Let be odd, then we prove that in order to describe any ideal of -graded identities of it is sufficient to deal with -grading, where , and if . In the same spirit of the case odd, if is even it is sufficient to study only those -gradings such that...