In my studies of physics and mathematics, I have encountered a fair bit of geometry, Lie group and representation theory, and real and complex analysis and I understand why these branches of mathematics are important. But I have learned very few applications of ring theory or other abstract algebra outside abstract algebra itself (save a few in number theory). At the same time, I believe it is considered vital for any aspiring mathematician to learn graduate level abstract algebra.
Why is abstract algebra considered to be so important? Examples of applications outside abstract algebra and outside mathematics would be appreciated.
To narrow the scope of the question down a bit, I am specifically asking about the theory of rings, fields, etc. I realize that the term 'abstract algebra' is a bit broader than what I intended.