Instytut Matematyczny Polskiej Akademii Nauk / Institute of Mathematics / Publishing house / Journals and Serials / Colloquium Mathematicum / All issues

Abstract

We introduce the abelian category $R$-gr of groupoid graded modules and give an answer to the following general question: If $U:R \hbox {-gr}\rightarrow R\hbox{-mod}$ 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 $X$ $\Rightarrow$ $U(M)$ has property $X$; (II) $U(M)$ has property $X$ $\Rightarrow$ $M$ has property $X$? We treat the cases when $X$ is one of the properties: direct summand, free, finitely generated, finitely presented, projective, injective, essential, small, and flat. We also investigate when exact sequences are pure in $R$-gr. Some relevant counterexamples are indicated.