Unit (ring theory)

https://arbital.com/p/unit_ring_theory

by Patrick Stevens Jul 28 2016

A unit in a ring is just an element with a multiplicative inverse.


[summary: A unit of a ring is an element with a multiplicative inverse.]

An element of a non-trivial ring%%note:That is, a ring in which ; equivalently, a ring with more than one element.%% is known as a unit if it has a multiplicative inverse: that is, if there is such that . (We specified that the ring be non-trivial. If the ring is trivial then and so the requirement is the same as ; this means is actually invertible in this ring, since its inverse is : we have .)

is never a unit, because is never equal to for any (since we specified that the ring be non-trivial).

If every nonzero element of a ring is a unit, then we say the ring is a field.

Note that if is a unit, then it has a unique inverse; the proof is an exercise. %%hidden(Proof): If , then (by multiplying both sides of by ) and so (by using ). %%

Examples