When studying abstract algebra, we might wonder: why do we always use “0” for the additive identity in rings? Is this just convention, or is there something deeper? Why don’t we choose a less opinionated notation like $i$ to distinguish the additive identity of a ring? This article proves there’s a mathematical inevitability to calling it zero.

To see why, let’s use the notation $i$ to denote the additive identity of a ring as opposed to the usual $0$, as suggested above. Where does it get us?

The necessity of rings when we have more than one operation

Our starting point for creating a group is a set $ G $ equipped with a single binary operation. What, however, limits us to only one operation on a group? Consider the integers: they have addition and multiplication defined on them. This motivates the need to extend the notion of a group to structures with more than one operation. A ring is introduced precisely to serve this purpose.

Have you heard the phrase “X is a markup language” where X could be HTML, Markdown, LaTeX, or XML? But what exactly is a markup language? Let’s do a thought experiment. Imagine that you’re a busy author who writes books but don’t want to have to format the book yourself. You don’t want to worry about the table of contents, the page numbers, the footnotes, the bold words… So you engage a publisher that will format your book properly.

Understanding Operations in Group Theory

What’s dark chocolate?

Dark chocolate is a form of chocolate made from…

Can you understand what dark chocolate is if you don’t know anything about chocolate? That’s the same thing for operations in group theory.

A group is a set equipped with a binary operation that satisfies four axioms. Like the definition of dark chocolate, the notion of operation is coupled with the definition of a group. Learn about operations and sharpen your knowledge of groups for free!