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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“Even in harmonious families there is this double life: the group life, which is the one we can observe in our neighbour’s household, and, underneath, another—secret and passionate and intense—which is the real life that stamps the faces and gives character to the voices of our friends. Always in his mind each member of these social units is escaping, running away, trying to break the net which circumstances and his own affections have woven about him.”
—Willa Cather (1873–1947)