Definition

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 \(y\cdot x=1\).
A golden ring floating with the word

Examples

  • Gaussian Integers

    The Gaussian integers are when you get when you add \(i\) to the integers. It's the ring \(\mathbb{Z}[i]\). The only invertible elements under multiplication are \(\{\pm 1,\pm i\}\).

  • Integers mod 6

    If you look at the integers mod 6, which we can think of as \(\{0,1,2,3,4,5\}\), then the only numbers with a multiplicative inverse are \{1, 5\}. The inverse of \(5\) is itself because: \[ 5\times 5\equiv 25 \equiv 1 \pmod{6}\]

  • \(2\times 2\) Real Matrices

    The multiplicative identity is the identity matrix \(I\). A real matrix will have an inverse if its determinant is not zero. So the units of this set are all matrices with non-zero determinants.

Group Theory

An Abstract Algebra Course by Socratica

Cayley Tables

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 \(\textit{group operation table}\).
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