# Why do the concepts of “zero divisor” and “unit” have so different names?

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)?

• Note that in general $x^n=0 \Rightarrow x=0$ is false. Elements such that $x^n=0$ are called nilpotents (also nontrivial nilpotents if $x \neq 0$). So, the concept of "roots of zero" does exist and is non-trivial. – lisyarus Oct 7 '18 at 13:05
• Thanks for the hint, I corrected it. – Hans-Peter Stricker Oct 7 '18 at 13:09
• You might also want to include "nilpotent" and "idempotent" in your catalogue. – Mark Bennet Oct 7 '18 at 13:12
• To express a purely personal view, I think it’s something of a waste of time to expect the names of mathematical phenomena to be reasonable or reasonably descriptive. Much better to call a unit “invertible”, but nobody does that. “Zero-divisor” actually is perfect: $a|b\Leftrightarrow\exists z$ with $za=b$, and $a$ is a zero-divisor if and only if $a|0$. But mathematics is riddled with unfortunate or undescriptive names: “good reduction”, “tropical geometry”. But you’re spitting (or worse) into the wind to try to change them. – Lubin Oct 7 '18 at 21:51
• WTo my mind, whether I like a term is not an issue, I fear. we’re stuck with them. There are worse things in mathematics, like $\sin^2$ in conflict with $\sin^{-1}$. – Lubin Oct 8 '18 at 15:41

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 $$1$$."

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 $$p = a\cdot b$$

• I think your first description of prime and irreducible is not nearly as valuable as the real definitions and does not reveal anything interesting. Indeed, I have a hard time seeing why you think the same thing with y's added is useful. What is the use of superficial similarity when the conceptual differences are buried? – rschwieb Oct 8 '18 at 13:14
• Sorry, but I have a hard time seeing what you mean: a) My definitions are exactly the real definitions, aren't they? (I point to the same Wikipedia article than you do. The "interesting" thing is, that the definitions are the same when replacing "unity" by "unit".) b) What is "the same thing" and what does "with y's added" mean? Which y's? c) In which sense are which conceptual differences "buried"? When there are no conceptual differences, why do you say the similarities on the terminological side are "superficial": they reflect the conceptual similarities. – Hans-Peter Stricker Oct 8 '18 at 13:37
• Anyway: I just discovered your marvelous DaRT - a beautiful resource! (My first question was: Why is there no entry for "unital" in the list of properties? But before I could ask, I discovered the FAQ - and now I know!) – Hans-Peter Stricker Oct 8 '18 at 13:43
• BTW: Do you know of any companion web sites to DaRT? DaAA (abstract algebra)? DaGrT (group theory)? DaGaT (Galois theory)? DaCT (category theory)? (In a way, The Stacks Project might count?) – Hans-Peter Stricker Oct 8 '18 at 13:49
• I finally found out: the DaRT-search yields examples, and it supports boolean NOT. – Hans-Peter Stricker Oct 8 '18 at 15:11