# nLab unit

Units

### Context

#### Algebra

higher algebra

universal algebra

## Theorems

#### Monoid theory

monoid theory in algebra:

# 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 real vector space. Since $\mathbb{R}$ is a field, any non-zero element 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.