Unit (ring theory)

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 $${\displaystyle vu=uv=1,}$$ 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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