Unit (ring theory)

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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. an element v such that

uv = vu = 1R, where 1R is the multiplicative identity.[1][2]

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 zero 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".)

The multiplicative identity 1R and its opposite −1R are always units. Hence, pairs of additive inverse elements[3] x and x are always associated.

Group of units[edit]

The units of 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) (for the German term Einheit).

In a commutative unital ring R, the group of units U(R) acts on R via multiplication. The orbits of this action are called sets of associates; in other words, there is an equivalence relation ∼ on R called associatedness such that

rs

means that there is a unit u with r = us.

One can check that U is a functor from the category of rings to the category of groups: every ring homomorphism f : RS induces a group homomorphism U(f) : U(R) → U(S), since f maps units to units. This functor has a left adjoint which is the integral group ring construction.

In an integral domain the cardinality of an equivalence class of associates is the same as that of U(R).

A ring R is a division ring if and only if U(R) = R \ {0} .

Examples[edit]

References[edit]

  1. ^ Dummit, David S.; Foote, Richard M. (2004). Abstract Algebra (3rd ed.). John Wiley & Sons. ISBN 0-471-43334-9. 
  2. ^ Lang, Serge (2002). Algebra. Graduate Texts in Mathematics. Springer. ISBN 0-387-95385-X. 
  3. ^ In a ring, the additive inverse of a non-zero element (1R or else) can equal to the element itself, in which case these statements become a vacuous truth.