Welcome to Ring Theory!
Ring Theory is an extension of Group Theory, vibrant, wide areas of current research in mathematics, computer science and mathematical/theoretical physics. They have many applications to the study of geometric objects, to topology and in many cases their links to other branches of algebra are quite well understood.
So, why study ring and group theory?
You might study them because you've got a research question that somehow involves symmetry --- constraint problems in computer science can be solved more efficiently when a little is known about the solution space. You might need to make some difficult calculations in a complicated topological space which become easier by passing into associated algebraic structures.
You might even want to make millions as a cryptographer --- current methods in cryptography are hard to crack, but become very easy when hacking with a quantum computer. There are good reasons to hope that rings and groups hold the key to codes that cannot be broken easily by quantum computers, just in case someone invents one and fancies stealing identities. We've just not come up with the perfect code yet.
Or perhaps simply because it's fun and not really difficult to get into. Ring theory can be understood at a moderate level by high-school level students, and in fact well enough by interested undergraduate students for them to produce original research. This is in stark contrast to the New Math introduced in American high schools in the mid-20th century, which consisted of teaching young teenagers abstract subjects like Set Theory and in certain cases Category Theory, both of which are very difficult to get an idea about without a solid grounding in undergraduate mathematics.
It's also a very beautiful subject. Many familiar mathematical objects are some sort of ring - some in more than one way. Understanding a little about rings can make it easier to understand these objects too.
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