In mathematics, an invertible element or a unit in a (unital) ring R is any element u that has an inverse element in the multiplicative monoid of R, i.e. an element v such that

uv = vu = 1_{R}, where 1_{R} 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.
The term unit is also used to refer to the identity element 1_{R} 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_{R} "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 1_{R} and its opposite −1_{R} are always units. Hence, pairs of additive inverse elements^{[3]} x and −x are always associated.
Group of units
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).
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

r ∼ s
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 : R → S 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
References

^ Dummit, David S.; Foote, Richard M. (2004). Abstract Algebra (3rd ed.).

^

^ In a ring, the additive inverse of a nonzero element can equal to the element itself.
This article was sourced from Creative Commons AttributionShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and USA.gov, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for USA.gov and content contributors is made possible from the U.S. Congress, EGovernment Act of 2002.
Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.
By using this site, you agree to the Terms of Use and Privacy Policy. World Heritage Encyclopedia™ is a registered trademark of the World Public Library Association, a nonprofit organization.