Why do the concepts of "zero divisor" and "unit" have so different names?

Question

I wonder why the concepts of zero divisor and unit have so different and unrelated names, even though their definitions are in perfect analogy:

$x$ is a zero divisor when there is a $y$ with $x\cdot y = 0$.

$x$ is a unit when there is a $y$ with $x\cdot y = 1$.

This means, "units" might be called "divisors of unity" but they aren't. Note that "divisor of unity" builds a conceptual bridge between other concepts:

$x$ is a divisor of zero.

$x$ is a divisor of unity ("unit").

$x$ is a root of unity.

$x$ is a root of zero ("nilpotent"). _{[Thanks to user lisyarus for the hint!]}

But since the mathematical community once decided to call divisors of unity "units", today everybody calls them units, but everybody knows that units actually are "divisors of unity" (they are only not named as such) - so there is not really a problem.

Except maybe for the beginners: From my own experience I can tell that for reading texts about ring theory it would have helped me if consequently "divisor of unity" would have been used instead of "unit".

My question is:

For which reasons did the mathematical community decide to call divisors of unity "units"?

I guess the term "unit" has been choosen because divisors of unity have a lot in common with unity itself:

• In $\mathbb{C}$ for example they have the same norm 1 as the unity.

• In integral domains (i.e. rings without zero divisors) one defines two elements to be associated when $a | b$ and $b | a$. Units are by definition exactly those elements that are associated with unity. But what about other kinds of rings?

• They are something like generalized unities. (So they would better be called "unitoids" but this sounds awkward, so one chose "unit"?)

What specifically do divisors of unity have in common with unity itself making them "unity-like"? And does this really justify to call them units instead of divisors of unity (what they actually are)?

Answer 1

The term "unit" meaning "the identity of the ring" undoubtedly comes from times when we thought about numbers as measuring things, and $1$ was the base unit with which other lengths were constructed. Personally I don't favor the use of "unit" to mean the multiplicative identity of a ring, since it is then confusingly also applied to invertible elements.

I would agree that an underlying connection of the two things you mentioned is divisbility.

Now, intimately related with the intuition for divisibility is factorization in $\mathbb Z$, and the idea that we shouldn't differentiate elements that generate the same ideal. For example, $(2)=(-2)$, and we don't care that $2\neq -2$ when it comes to divisibility. That is the reason we have the notion of associates, which are, in integral domains, elements that generate the same principal ideal.

In particular, $(1)=(u)$ for any "unit"/"divisor of $1$" $u$. So the thing units have in common is that they all generate the same ideal as $1$.

So in summary, I would say that "units" and "the unit" were tied together as people developed and taught the theory of divisibility in integral domains like $\mathbb Z$, and in that context all units "behave the same as $1$."

Answer 2

A consequence of calling divisors of unity "units" is that the two definitions of prime number and irreducibles elements come in analogy.

$p \in \mathbb{N}$ is a prime number if

• $p$ is not zero or unity

• there are no $a,b$ both not being unity with $p = a\cdot b$

Let $\mathcal{R}$ be a integral domain.

$p \in \mathcal{R}$ is a irreducible element if

• $p$ is not zero or a unit

• there are no $a,b$ both not being units with $p = a\cdot b$

Being able to make these definitions in analogy might have been one reason for calling divisors of unity "units".

For $\mathcal{R} = \mathbb{Z}$:

$p \in \mathbb{N}$ is a prime number if

• $p \neq 0,1$

• there are no $a,b \neq 0,1$ with $p = a\cdot b$

$p \in \mathbb{Z}$ is a irreducible number if

• $p \neq 0,\pm 1$

• there are no $a,b \neq 0,\pm 1$ with $p = a\cdot b$