nLab
unit

For special kinds of units see also unit of an adjunction and unit of a monad .


Units

Idea

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

Definitions

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

Given sets R and M, and a function :R×MM, an element u of M is a unit (relative to the operation ) if, given any element x of M, there exists a unique element a of R such that x=au.

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

Units in rings

If R is a ring (or rig), then R comes equipped with a multiplication map :R×RR. 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=(xu 1)ux = (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=au1 = a u

(on the left) or

1=ua1 = 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=au;x = a u ;

and u is a right unit if, for every x, there is an a 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 R is a ring (or rig) and M an R-module, then a unit in M is an element uM such that every other xM can be written as x=au (or x=ua for a right module) for some aR. 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 -line, a 1-dimensional vector space over the real numbers. Since 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 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 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 0-module) and may be expressed as a nonnegative real multiple of the kilogramme.

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 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)