unit

For special kinds of units see also

unit of an adjunctionandunit of a monad.

A unit is a quantity $u$ such that every other quantity (of a certain type) is a multiple (in a certain sense) of $u$.

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

Given sets $R$ and $M$, and a function $\cdot :R\times M\to M$, 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 $.

If $R$ is a ring (or rig), then $R$ comes equipped with a multiplication map $\cdot :R\times R\to R$. 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

$$1=au$$

(on the left) or

$$1=ua$$

(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.

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.

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

$$x=au;$$

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

$$x=ua.$$

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.

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 $x=au$ (or $x=ua$ 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.

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 $R\u2254\mathbb{R}$ 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 kilogramme is a unit of mass, because any mass may be expressed as a real multiple of the kilogramme. 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 kilogramme.

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 ring with a unit must have an identity.

I haven't managed to find either a proof or a counterexample to the converse: 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=au, 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 $bu$, and then $av=a(bu)=b(au)=bu=v$. But without commutativity (and associativity), this doesn't work.

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

Revised on February 15, 2013 12:59:21
by Toby Bartels
(98.23.143.39)