In the same way “integers” are a set of objects, “numbers” have rules about how they interact. For example “addition” or “multiplication” of two “numbers”. These rules on how they interact are called “axioms” and they tell us about the nature and how these groups of numbers interact. Rings are similiar in terms of having elements or sybols that we defined and then add or subtract based on specific rules.
“Integers have lots of properties that continue to work if we scale back on the number of “rules” we are willing to take for granted? What can we still prove? For every set of “rules” we take, we can develop an entire theory of results which require nothing more than the set of rules we started out with. For one specific set of rules, any object which is found satisfying all of those properties is called a ring (though people differ slightly on exactly what they call a ring).
There is a different set of rules which define what people call a group (the definition of a group is more set than that of a ring), a different one still for a vector space, one for a field, … The list goes on. People have dreamed up lots of different sets of rules which, for some reason or another, are more relevant to what they find interesting. More generally, this whole idea is the idea of abstraction; a ring is just one of many abstract objects.
On the other hand, when you strip many of the extra details out and just focus on these smaller sets of rules, some facts become more obviously true, because it is clear which properties imply the result you are looking for.
Rings are objects with sufficiently general rules as to gain insight on objects from the set of integers or the set of complex numbers, to the set of 3×3 matrices, or the set of all functions from the real numbers to the real numbers.
These are at least the thoughts that went into the formation of the theory (though they don’t much distinguish between, say, ring theory and group theory). A bit more specifically, the notion of a ring is a generalization of standard number systems, in that they retain two distinct operations with “multiplication” distributing over “addition” (though a general ring does not require that multiplication commutes as is the case for matrices). An exact definition of a ring is easy enough to put down, but not particularly enlightening without more context. Hopefully this helps provide some of the context. ” – Source: mymathforum.com/abstract-algebra/20100-simplified-explanation-algebraic-ring-theory.html
