nLab unit




Monoid theory

For other kinds of units see also unit of an adjunction and unit of a monad. Different (but related) is physical unit.



Considering a ring RR, then by the unit element or the multiplicative unit one usually means the neutral element 1R1 \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.


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

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

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

Units in rings

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

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

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

(on the left) or

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

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

1=au 1 = a u

(on the left) or

1=ua 1 = u a

(on the right).

In a commutative ring (or rig), a unit is an element of RR that has an inverse, period. Of course, a commutative ring RR 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; uu is a left unit if, for every xx, there is an aa such that

x=au; x = a u ;

and uu is a right unit if, for every xx, there is an aa such that

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

Units of measurement

In physics, the quantities of a given dimension generally form an \mathbb{R}-line, a 11-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 RR \coloneqq \mathbb{R} and MM 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 0\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 0\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 11. 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 11 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 RR is a rng with a unit uu, then every element uniquely factors through uu. In particular, uu itself does. u=auu = a u, with aa unique. So aa is an identity.

Reply: Why is aa an identity then? This works if the rng is commutative: given any vv, write vv as bub u, and then av=a(bu)=b(au)=bu=va 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.


See also

Last revised on August 21, 2023 at 13:20:35. See the history of this page for a list of all contributions to it.