The existence of identity of orthodox semigroup rings.
The existence of relative pure injective envelopes
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.
The Exterior Derivation and the Orders of Differential Forms on Contractible Algebras.
The Extraordinary Derived Category.
The finite dimension property of the irreducible representations of Noetherian -Lie algebras.
The flat dimensions of injective modules.
The free -ring is a graded -ring.
The Frobenius theorem on -dimensional quantum hyperplane.
The fundamental theorem and Maschke's theorem in the category of relative Hom-Hopf modules
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.
The -graded identities of the Grassmann Algebra
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...
The Galois algebras and the Azumaya Galois extensions.
The Galois extensions induced by idempotents in a Galois algebra.
The Galois module structure of algebraic integer rings in fields with generalised quaternion group
The Gelfand-Kirillov Dimension of a Tensor Product.
The Gelfand-Kirillov dimension of rings with Hopf algebra action.
The general Ikehata theorem for -separable crossed products.
The general structure of inverse polynomial modules
In this paper we compute injective, projective and flat dimensions of inverse polynomial modules as -modules. We also generalize Hom and Ext functors of inverse polynomial modules to any submonoid but we show Tor functor of inverse polynomial modules can be generalized only for a symmetric submonoid.
The geometric reductivity of the quantum group
We introduce the concept of geometrically reductive quantum group which is a generalization of the Mumford definition of geometrically reductive algebraic group. We prove that if G is a geometrically reductive quantum group and acts rationally on a commutative and finitely generated algebra A, then the algebra of invariants is finitely generated. We also prove that in characteristic 0 a quantum group G is geometrically reductive if and only if every rational G-module is semisimple, and that in...
The Geometry of Representations of Am .