Often times in my studies I get the impression that algebra is just a tool to help with other branches of mathematics, like algebraic geometry, algebraic number theory, algebraic topology, etc. How true this is, I am not sure.
So I suppose I want to ask, what sort of work do modern day algebraists do?
What are currently some of the more active areas of modern algebra?
What types of problems do algebraists deal with?
I'm kicking around the idea of pursuing graduate study one day, possibly in some sort of algebraic field, i.e., ring theory or something. What sort of research and problems are open to your average graduate student in algebra (of any sort, not just ring theory)?
This is partially inspired by the question What do modern-day analysts actually do?
Thank you for your responses.