Some Properties of a Subring of the Power Series Ring on a Countably Innnite Number of Variables over a Eld

We prove some simple properties of the power series ring R = Kx 1 ; x 2 ; x 3 ; : : : ]] on a countably innnite number of variables over a eld K, and of the subring R 0 generated by all homogeneous elements in R. By means of a certain decreasing ltration of ideals, which are kernels of the \truncation homomorphisms" n : R 0 ! Kx 1 ; : : : ; x n ], we endow R 0 with a topology, and show that with respect to this topology, homogeneous, nitely generated ideals are closed. We also show that the truncation homomorphisms \commute in a ltered sense" with the formation of greatest common divisors (and least common multiples): for any homogeneous f; g 2 R 0 , there exists an N such that for n > N , gcd(n (f); n (g)) = n (gcd(f; g)).