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The category of groupoid graded modules

Patrik Lundström (2004)

Colloquium Mathematicae

We introduce the abelian category R-gr of groupoid graded modules and give an answer to the following general question: If U: R-gr → R-mod denotes the functor which associates to any graded left R-module M the underlying ungraded structure U(M), when does either of the following two implications hold: (I) M has property X ⇒ U(M) has property X; (II) U(M) has property X ⇒ M has property X? We treat the cases when X is one of the properties: direct summand, free, finitely generated, finitely presented,...

The G -graded identities of the Grassmann Algebra

Lucio Centrone (2016)

Archivum Mathematicum

Let G be a finite abelian group with identity element 1 G and L = g G L g be an infinite dimensional G -homogeneous vector space over a field of characteristic 0 . Let E = E ( L ) be the Grassmann algebra generated by L . It follows that E is a G -graded algebra. Let | G | be odd, then we prove that in order to describe any ideal of G -graded identities of E it is sufficient to deal with G ' -grading, where | G ' | | G | , dim F L 1 G ' = and dim F L g ' < if g ' 1 G ' . In the same spirit of the case | G | odd, if | G | is even it is sufficient to study only those G -gradings such that...

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