ON GRADED RINGS WITH FINITENESS CONDITIONS

It is proved that a graded ring that is finitely graded modulo its radical is local if its initial subring is local, and that a graded artinian ring is finitely generated over its initial subring which is also artinian. These results extend theorems of Gordon and Green on artin algebras. Other results relating the structure of a graded ring to that of its initial subring are also presented. In this note we provide simple proofs of extensions of the principal results in R. Gordon and E. Green's recent paper (4) on graded artin algebras. First we prove that any finitely graded (modulo the radical) ring with local initial subring is itself local. From this it follows that over any graded ring a finitely graded module with a composition series is an indecomposable module if and only if it is indecomposable as a graded module. Then we prove that a graded ring is left artinian if and only if it has a composition series as a left module over its initial subring. These results are used to show that over a graded left artinian ring every simple module, every projective module, and every injective left module is isomorphic to a graded module, as is every direct summand of a finitely generated graded left module. Also they yield information about the relative structure of a graded ring and its initial subring, for example, a finitely graded ring is semiprimary if and only if so is its initial