After reading some papers about special kinds of algebras and rings like Gorenstein rings, Dickson algebras, Cayley-Dickson construction, i want to ask do examples of general abstract objects in math like groups, rings, manifolds, abstract curvatures, modular forms that help us understand helps us know where the directions of math go to?
For example, the n-manifold in n dimentions is defined and as special cases we have curves and surfaces, another example is the groups , where we have the set of real numbers equipped with addition and gives us the abelian group $(R,+)$, or with the complex numbers. Then we learn other examples of groups like the group of matrices equipped with addition or multiplication where the first is abelian group and the second nonabelian respectively.
Questions: Does this way help us understand generally things in math?By considering special cases? Do you know cases where this perhaps is not applied? If it is not applied could it be applied by analogy?If i try to read math by considering special cases of objects to help me is this valid and okay?