Thanks Fernando! That is a nice and elementary example. I was looking in the wrong place.

@Jonny: You are right, and the answer to Alan's question "Is it acceptable...?" is of course "Yes!". The purpose of my comment was to demonstrate that change is usually considered to be a healthy and good thing.

Not only is it desirable to broaden your research after completing your PhD studies. By some national funding agencies, such as the Swedish Research Council for example, this is actually a requirement to be elegible for their postdoctoral scholarships. You need to present a research plan that shows that you are are not "getting stuck" in old tracks.

Nice answer! Just a curiosity... Huishi Li, whose paper you refer to, is the same guy that wrote the book "Zariskian Filtrations" (K-Monographs in Mathematics) together with Van Oystaeyen.

Have you consulted the book "Methods of graded rings" (Springer Lecture Notes in Mathematics) by Constantin Nastasescu and Fred Van Oystaeyen? Chapter 2.9 deals with the "graded Jacobson radical" and in Corollary 2.9.3 they show that $J^g(R) \cap R_0 = J(R_0)$ (in fact they show this for gradations by arbitrary groups). I don't know if this is of any help to you though, but I thought I'd better mention it.

This is similar to Lars' comment above. On the bottom floor of the math building at Lund University (Sweden) you'll find a students' cafe named "Hilbertrummet" (which means both "the Hilbert room" and "the Hilbert space").