What motivates the definition of a ring in abstract algebra?

Question

I've read a lot of questions about rings but I still don't know why they are useful/ I'm sure they are, but I don't know why. Are their properties somehow used in proofs or as foundations for theorems? And what exactly do mathematicians study when they research rings? Can you give some examples?

Answer 1

Although I am not much of an Algebraic Geometry person, but it is a very important application of commutative ring theory. Just to give you a flavor, here is a brief idea.

Consider the ring $\Bbb{C}$. Now we do like to study surfaces/curves (that is an application, right!) , and we define them by equations in rings $R=\Bbb{C}[x_1,x_2,\dots , x_n]$.

Now suppose $p(x)\in R$ is a polynomial, and we need to find zeroes of this polynomial (for same reasons we find zeros of quadratics in single variable, they help us study surfaces and many things), then it is not as straightforward as a quadratic\cubic equation in $\Bbb{C}[x]$.
We need the locus $X=V(p(x))=\{t=(a_1,a_2,\dots , a_n): p(t)=0\}\subseteq \Bbb{C^n}$

Now consider the residue ring $A=\Bbb{C}[x_1,x_2,\dots , x_n]/(p(x))$. Now a proposition in algebraic geometry (in beginning of it) says Maximal ideals of $A$ are in one-one correspondence with points $P\in X$. So studying maximal ideals of these rings gives us solution of those multivariable polynomials.

For Example, Consider the Curve $x^3-y^2 \in \Bbb{C}[x,y]$.

Now to find set of complex 2-tuples such that they satisfy $x^3-y^2=0$ (i.e. solutions), we can correspond them with maximal ideals of the ring $A=\Bbb{C}[x,y]/(x^3-y^2)$ , and $A$ can be realized as the same ring $A'=\Bbb{C}[x][\sqrt{x^3}]$ and by Hilbert Nullstellensatz, maximal ideals of $A'$ are of the form $(x-a,y-b)$ where $b=\pm\sqrt{a^3}$, $a,b\in \Bbb{C}$ .

Also clearly $A'=\Bbb{C}[x][\sqrt{x^3}] \subset \Bbb{C}[\sqrt{x}]$, then letting $t=\sqrt{x}$, we see $A' \subset \Bbb{C}[t]$, where $\Bbb{C}[t]$ is a PID and all its ideals are maximals, must be useful, though I am not sure how exactly.

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Similarly local rings (rings with a unique maximal ideal) are of utmost importance in algebraic geometry, and we can get a local ring from any ring via Localization process and then study curves/surfaces over them is useful etc.

Study of ideals of rings, finding all the prime ideals and maximal ideals of various rings helps us in many branches of maths, esp. in Algebraic geometry.

It is just one of the many important things, rings offer us. Just do not forget all fields are rings, and fields are important to us as from the very beginning we liked working with real and complex number, which both are fields and therefore rings.

We use Galois theory to prove every quintic is unsolvable by radicals etc. Rings play important part in algebraic number theory and helped proving as big and important conjectures as Fermat's Last theorem.