My Professor of Homological Algebra got me into some Hochschild (co)homology and then suggested to continue with formally smooth algebras, noncommutative differential forms and so forth.
Now, my personal interest has always been algebra, especially noncommutative (ring and module theory, category theory, some homological algebra). Moreover, my studies have been concerned almost exclusively with such subjects. Hence, I am very bad at calculus, differential geometry etc. I do have the basics, though. I know it's recommendable to have a decent knowledge in most of the subjects, but I got far enough in algebra not needing very much knowledge of calculus, say.
What I want to ask you is the following: Assuming that I got on the way to noncommutative geometry (so it seems to me...), how much algebra is there in? As I said, I really want a career in noncommutative algebra and it would be pretty unpleasant to get involved intensively in a subject which is not my cup of tea. So far, so good, I am enjoying the subject and it interests me in a personal way (not just for school), but I am only at the beginning, I have met only the most basics.
I read that Connes somehow started the subject of NG wanting to extend differential geometry for arbitrary (noncommutative) rings, but would you say the current research work in NG, cyclic homology and the like is algebra at its finest or does it have deep and links with something else (what?)?
OR am I getting this wrong and there are some other paths ahead, starting from noncommutative differential forms, formally smooth algebras, Hochschild (co)homology and the like?