Unit (ring theory)
Less commonly, the term unit is also used to refer to the element 1 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 1 "unity" or "identity", and say that R is a "ring with unity" or a "ring with identity" rather than a "ring with a unit".
in the ring, so √5 + 2 is a unit. (In fact, the unit group of this ring is infinite.)
In fact, Dirichlet's unit theorem describes the structure of U(R) precisely: it is isomorphic to a group of the form
For a commutative ring R, the units of the polynomial ring R[x] are precisely those polynomials
The unit group of the ring Mn(R) of n × n matrices over a ring R is the group GLn(R) of invertible matrices. For a commutative ring R, an element A of Mn(R) is invertible if and only if the determinant of A is invertible in R. In that case, A−1 is explicitly given by Cramer's rule.
The units of a ring R form a group U(R) under multiplication, the group of units of R.
Other common notations for U(R) are R∗, R×, and E(R) (from the German term Einheit).
Suppose that R is commutative. Elements r and s of R are called associate if there exists a unit u in R such that r = us; then write r ∼ s. In any ring, pairs of additive inverse elements[a] x and −x are associate. For example, 6 and −6 are associate in Z. In general, ~ is an equivalence relation on R.