## Units in a Ring

The Invertible Parts of a Ring (for multiplication)

Units in a Ring

Rings have two operations: \(+\) and \(\times\). \[\]
Any ring \(R\) is a commutative group under addition, so every element has an additive inverse. But multiplication is different. In general there will be lots of elements without an inverse under \(\times\). If an element \(x\in R\) has an inverse we call it a \(\textit{unit}\). The set of all units is denoted by \(R^\times\). \[\]
So if \(x\in R\) is a unit, then there is some element \(y\in R\) so that \(x\cdot y=1\) and \(y\cdot x=1\).

An Abstract Algebra Course by Socratica

A related lesson

A Cayley table is a group multiplication table. We say "multiplication" as short-hand. To be precise a Cayley table is an \(\textit{group operation table}\).