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How do mathematicians study rings and groups?

Rings and groups are a set of elements closed under an operation (or something like that). What can mathematicians study from these elements in rings and groups? What are some of the applications? Thank you.

Public Comments

1. They study the properties of rings and groups rather than the elements themselves. As for applications, I didn't see much of it in my career as a student.
2. There are many applications of group theory. See the Wikipedia article, link below. About rings I am not so sure. However the next step up from a ring is a field, and here there are many applications as well. Disclaimer: I don't know all that much algebra, and an expert might give you a very different answer.