Groups, Rings, And Fields: Algebra:

The order of grouping doesn't change the result.

Every element has an opposite that brings it back to the identity. Algebra: Groups, rings, and fields

Fields are essential for solving equations. Because every non-zero element has a multiplicative inverse, we can isolate variables and find exact solutions. They are the backbone of linear algebra and most physics simulations. The order of grouping doesn't change the result