Unit (ring Theory)
In mathematics, an invertible element or a unit in a (unital) ring R refers to any element u that has an inverse element in the multiplicative monoid of R, i.e. such element v that
- uv = vu = 1R, where 1R is the multiplicative identity element.
The set of units of any ring is closed under multiplication (the product of two units is again a unit), and forms a group for this operation. It never contains the element 0 (except in the case of the trivial ring), and is therefore not closed under addition; its complement however might be a group under addition, which happens if and only if the ring is a local ring.
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" or "identity", and say that R is a "ring with unity" or a "ring with identity" rather than a "ring with a unit".)
Read more about Unit (ring Theory): Group of Units, Examples
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