Abstract
Introduction Although this note is a sequel of our paper [7], the technique we use now is completely different, being very much inspired by the one we have used in the proof of [S, Theorem 2.21 (see the key result, Proposition 2.1). In [7] we have studied the following problem: does a graded module with graded Gabriel dimension have a Gabriel dimension when regarded without grading? We gave in [7] a positive answer to this problem in the commutative case, as well as a relation between the two dimensions, leaving the problem open in the non-commutative case. The purpose of this note is to settle the problem completely, and to use the same (adapted) simple argument in order to obtain an evaluation for the Gabriel dimension of polynomials, previously proved by Gordon and Robson [5] using polynomial categories. The reader may note that our proof avoids the use of quotient categories and polynomial categories. 1. Notation and preliminaries All rings considered in this paper will be unitary. If
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