In algebra, a unit or invertible element[a] of a ring is an invertible element for the multiplication of the ring. That is, an element u of a ring R is a unit if there exists v in R such that where 1 is the multiplicative identity; the element v is unique for this property and is called the multiplicative inverse of u.[1][2] The set of units of R forms a group R× under multiplication, called the group of units or unit group of R.[b] Other notations for the unit group are R∗, U(R), and E(R) (from the German term Einheit).
Less commonly, the term unit is sometimes used to refer to the element 1 of the ring, in expressions like ring with a unit or unit ring, and also unit matrix. Because of this ambiguity, 1 is more commonly called the "unity" or the "identity" of the ring, and the phrases "ring with unity" or a "ring with identity" may be used to emphasize that one is considering a ring instead of a rng.
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