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
- 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}.
Read more about this topic: Unit (ring Theory)
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