Ring Theory

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

Group theory

Math and Algebra study Algebraic groups. These are concepts in abstract algebra like ringsfields, and vector spaces, are all forms or algebraic groups.

We can see algrebraic groups in nature, like crystals and the hydrogen atom, in other words, we can translate them to math through socmeting called symmetry groups. This is why group theory is important to create mathematical models of phisical things like chemistry, and materials science. Group theory is also central to public key cryptography. But I´ll let the experts explain this they way experts know how.

What is Algebra?

“Algebra (from Arabic “al-jabr” meaning “reunion of broken parts”) is one of the broad parts of mathematics, together with number theory, geometry, and analysis. In its most general form, algebra is the study of mathematical symbols and the rules for manipulating these symbols; it is a unifying thread of almost all of mathematics. As such, it includes everything from elementary equation solving to the study of abstractions such as groups, rings, and fields. The more basic parts of algebra are called elementary algebra; the more abstract parts are called abstract algebra or modern algebra. Elementary algebra is generally considered to be essential for any study of mathematics, science, or engineering, as well as such applications as medicine and economics. Abstract algebra is a major area in advanced mathematics, studied primarily by professional mathematicians.

Elementary algebra differs from arithmetic in the use of abstractions, such as using letters to stand for numbers that are either unknown or allowed to take on many values. For example, in {\displaystyle x+2=5} the letter {\displaystyle x} is unknown, but the law of inverses can be used to discover its value: {\displaystyle x=3}. In E = mc2, the letters {\displaystyle E} and {\displaystyle m} are variables, and the letter {\displaystyle c} is a constant, the speed of light in a vacuum. Algebra gives methods for solving equations and expressing formulas that are much easier (for those who know how to use them) than the older method of writing everything out in words. The word algebra is also used in certain specialized ways. A special kind of mathematical object in abstract algebra is called an “algebra”, and the word is used, for example, in the phrases linear algebra and algebraic topology.” https://en.wikipedia.org/wiki/Algebra