We start from two sets A and B

Let $A$ and $B$ be two sets. A function is a mapping from elements of $A$ onto elements in $B$ that follows three rules:

The three fundamental rules for a function $f: A \to B$:

1. Domain Rule: For every element $x \in A$, there exists at least one element $y \in B$ such that $f(x) = y$.

2. Uniqueness Rule: For every element $x \in A$, there exists at most one element $y \in B$ such that $f(x) = y$.

Two friends can race at the exact same speed, yet one may be accelerating while the other is not. Why? The key lies in the velocity vector.

This idea is mathematically subtle, but it can be captured with the right intuition.

Speed is a scalar quantity. It tells you how fast something moves, but not where it is going. Velocity, on the other hand, is a vector. It has both a magnitude, which is the speed, and a direction.

You’re a girl going abroad for university. A friend tells you: a male classmate from back home will go to the same country. A coincidence, right?

Then you hear he’s in the same city. Closer.

A friend mentions his neighborhood—you tense—he’s moving to the same area.

Finally, you learn from your landlord that you’ll have a roommate. Guess who?

There’s no place for chance anymore. Let the symbol $\epsilon$ represents the distance you look around to check for the presence of your classmate. For every possible $\epsilon$ distance you look around, no matter how small, that stalker is always there. When $\epsilon$ was the size of a country, the classmate was there. $\epsilon$ became a city, a neighborhood, even as small as a house. But you could always find that stalker around. A limit point of a set behaves almost exactly like your situation with this classmate.

As a recap, the definition of a limit point of a set states that:

A point $x\in\mathbb{R}$ is a limit point of a set $A\subset\mathbb{R}$ if

$$ \forall \varepsilon>0\ \exists y\in A\setminus{x}\quad\text{such that}\quad |y-x|<\varepsilon. $$

Now, consider a sequence of real numbers:

$$ a_1, a_2, a_3, a_4, a_5, a_6, a_7, a_8, a_9, a_{10}, \dots $$

Adapting the set definition, one might naively say: for all $\varepsilon>0$, there exists an index $i$ such that $a_i$ is very close to the proposed limit point. Formally, the naive definition would be:

In everyday mathematics, zero and one are fundamentally different.

But abstract algebra asks: what if we relax our assumptions? What happens in a ring where the additive identity and the multiplicative identity are the same? Can such a ring even exist?

The answer reveals a simple but powerful fact: if $0 = 1$ in a ring, then the ring collapses to a single element.

The Setup

A ring $R$ possesses two binary operations, commonly denoted by $+$ and $\cdot$. Assuming we are working with a ring with unity (see ring with multiplicative identity), each operation has its own identity:

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?