The notion of graded ring is most frequently used when G is the semigroup {@I, 2,. . .} o non-negative integers, which we denote by lW. In f this case, we set R ’ = In > 0 R,; then R+ is an ideal of R, with R/R+ N R,. The polynomial ring R = R, [X,, . , X,] over a ring R, is usually made into an N-graded ring by defining the degree of a monomial Xt’ . . X2 as the total degree a, + ... + a,; however, R has other useful gradings. For example, R has an lk4”-grading in which X;l ...Xz has degree (a,, . . . , a,); the value of systematically using this grading can be seen in Goto- Watanabe [l]. Alternatively, giving each of the Xi some suitable weight di and letting the monomial Xb;l . ..X. have weight Iaid, defines an N-grading of R. For example, the ring R,[X, Y, Z]/(f), where f = aiXa +