Unit (ring theory)

In mathematics, a unit in a (unital) ring R is an invertible element of u, i.e. an element u such that there is a v in R with

Related Topics:
Mathematics - Unital - Ring

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

:uv = vu = 1R, where 1R is the multiplicative identity element.

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

That is, u is an invertible element of the multiplicative monoid of R.

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

Unfortunately, the term unit is also used to refer to the identity element 1R of the ring, in expressions like ring with a unit or unit ring, and also e.g. unit matrix. (For this reason, some authors call 1R "unity", and say that R is a "ring with unity" rather than "ring with a unit".)

~ ~ ~ ~ ~ ~ ~ ~ ~ ~