Are there any applications of abstract algebra (group theory, rings, fields etc.) in any branch of either economics or finance?
Yes, I've seen some interesting papers that improve one's insight into how things work, even if it is not clearly applicable to practice.
Belal Ehsan Baaquie published several books on applications of quantum mechanics and quantum field theory to finance, particularly interest rates. They're definitely fun to read. The most recent one is Quantum Field Theory for Economics and Finance (2018).
There was a related question here (not answered) a few years ago about using Lie Groups for interest rates. The paper Park, Chun, Han Webber, Interest rate models on Lie groups is an example of this.
There have been attemps to view credit from the category theory viewpoint, for example Joseph Tanega. Default Invariance, A Naïve Category Theory of Law and Finance.
Not quite finance, but Algebraic Models for Accounting Systems by Nehmer, Perez, Robinson, Rambaud explores a group-theoretic view of double-entry accounting.
Not quite modern algrebra, but related: