On graded torsion free and graded injective k[x] modules

Much information about rings can be gained by studying their modules, and similarly, graded rings can be studied by studying their graded modules. And just as modules can be studied by considering their various envelopes and coverings, graded modules can be studied using the graded counterpart of these notions. Graded torsion and graded torsion free modules often arise in the study of projective geometry. Artin and Zhang [2] use graded torsion and graded torsion free modules in their discussion of noncommutative projective varieties. It is known that a module over an integral domain has a unique torsion free covering [1]. In this paper, we initiate the study of graded torsion free, graded divisible, and graded injective modules over the (graded) integral domain k[x] (where k is a field).