Unit (ring theory)

In mathematics, a unit in a (unital) ring R is an invertible element of R, i.e. an element u such that there is a v in R withuv = 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". Note also that the term unit matrix more usually denotes a matrix with all elements equal to one.)
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