Units

Idea

Considering a ring $R$, then by the unit element or the multiplicative unit one usually means the neutral element $1 \in R$ with respect to multiplication. This is the sense of “unit” in terms such as nonunital ring.

But more generally a unit element in a unital (!) ring is any element that has an inverse element under multiplication.

This concept generalizes beyond rings, and this is what is discussed in the following.

Definitions

Exactly what this means depends on context. A very general definition is this:

Given sets $R$ and $M$, and a function ${\cdot}\colon R \times M \to M$, an element $u$ of $M$ is a unit (relative to the operation ${\cdot}$) if, given any element $x$ of $M$, there exists a unique element $a$ of $R$ such that $x = a \cdot u$.

That is, every element of $M$ is a multiple (in a unique way) of $u$, where ‘multiple’ is defined in terms of the operation ${\cdot}$.

Units in rings

If $R$ is a ring (or rig), then $R$ comes equipped with a multiplication map ${\cdot}\colon R \times R \to R$. So $R$ can play the role of both $R$ and $M$ above, although there are two ways to do this: on the left and on the right.

We find that $u$ is a left unit if and only if $u$ has a left inverse, and $u$ is a right unit if and only if $u$ has a right inverse. First, an element $u$ with an inverse is a unit because, given any element $x$, we have

$x = (x u^{-1}) u$

(on the left) or

$x = u (u^{-1} x)$

(on the right). Conversely, a unit must have an inverse, since there must a solution to

$1 = a u$

(on the left) or

$1 = u a$

(on the right).

In a commutative ring (or rig), a unit is an element of $R$ that has an inverse, period. Of course, a commutative ring $R$ is a field just when every non-zero element is a unit.

Units in monoids

Notice that addition plays no role in the characterisation above of a unit in a ring. Accordingly, a unit in a monoid may be defined in precisely the same way.

A group is precisely a monoid in which every element is a unit.

Units in rngs or semigroups

In a rng (or, ignoring addition, in a semigroup), we cannot speak of inverses of elements. However, we can still talk about units; $u$ is a left unit if, for every $x$, there is an $a$ such that

$x = a u ;$

and $u$ is a right unit if, for every $x$, there is an $a$ such that

$x = u a .$

Units in nonassocative rings or magmas

In a nonassociative ring (or, ignoring addition, in a magma), even if we have an identity element, an invertible element might not be a unit. So we must use the same explicit definition as in a rng (or semigroup) above.

A quasigroup is precisely a magma in which every element is a two-sided unit.

Units in modules

If $R$ is a ring (or rig) and $M$ an $R$-module, then a unit in $M$ is an element $u \in M$ such that every other $x \in M$ can be written as $x = a u$ (or $x = u a$ for a right module) for some $a \in R$. This is the same as a generator of $M$ as an $R$-module. There is no need to distinguish left and right units unless $M$ is a bimodule. Note that a (left or right) unit in $R$ qua ring is the same as a unit in $R$ qua (left or right) $R$-module.

Units of measurement

In physics, the quantities of a given dimension generally form an $\mathbb{R}$-line, a $1$-dimensional vector space over the real numbers. Since $\mathbb{R}$ is a field, any non-zero quantity is a unit, called in this context a unit of measurement. This is actually a special case of a unit in a module, where $R \coloneqq \mathbb{R}$ and $M$ is the line in question.

Often (but not always) these quantities form an oriented line, so that nonzero quantities are either positive or negative. Then we usually also require a unit of measurement to be positive. In fact, for some dimensions, there is no physical meaning to a negative quantity, in which case the quantities actually form a module over the rig $\mathbb{R}_{\ge 0}$ and every nonzero element is “positive.”

For example, the kilogram is a unit of mass, because any mass may be expressed as a real multiple of the kilogram. Further, it is a positive unit; the mass of any physical object is a nonnegative quantity (so that mass quantities actually form an $\mathbb{R}_{\ge 0}$-module) and may be expressed as a nonnegative real multiple of the kilogram.

Identities as units

Often the term ‘unit’ (or ‘unity’) is used as a synonym for ‘identity element’, especially when this identity element is denoted $1$. For example, a ‘ring with unit’ (or ‘ring with unity’) is a ring with an identity (used by authors who say ‘ring’ for a rng). Of course, a rng with identity has a unit, since $1$ itself is a unit; conversely, a commutative rng with a unit must have an identity.

I haven't managed to find either a proof or a counterexample to the converse (in the noncommutative case): that a rng with a unit must have an identity.

Response: If $R$ is a rng with a unit $u$, then every element uniquely factors through $u$. In particular, $u$ itself does. $u = a u$, with $a$ unique. So $a$ is an identity.

Reply: Why is $a$ an identity then? This works if the rng is commutative: given any $v$, write $v$ as $b u$, and then $a v = a (b u) = b (a u) = b u = v$. But without commutativity (and associativity), this doesn't work.

It is this meaning of ‘unit’ which gives rise to the unit of an adjunction.