Klein 4-ring

One of the two smallest non-commutative rings is the Klein 4-ring $(R,+,\cdot)$ where $(R,+)$ is the Klein 4-group $\{0,\,a,\,b,\,c\}$ with $0$ the neutral element and the binary operation “ $\cdot$ ” given by the table

$\begin{array}[]{c|cccc}\cdot&0&a&b&c\\ \hline\;0&0&0&0&0\\ \;a&0&a&0&a\\ \;b&0&b&0&b\\ \;c&0&c&0&c\end{array}$

Note that this ring has two different right unities $a$ and $c$ .

The Klein 4-ring has the subrings $\{0,\,a\}$ , $\{0,\,b\}$ and $\{0,\,c\}$ and the two-sided ideal $\{0,\,b\}$ .

Title	Klein 4-ring
Canonical name	Klein4ring
Date of creation	2015-06-04 16:11:18
Last modified on	2015-06-04 16:11:18
Owner	pahio (2872)
Last modified by	pahio (2872)
Numerical id	16
Author	pahio (2872)
Entry type	Definition
Classification	msc 20-00
Classification	msc 16B99
Synonym	Klein’s four-ring
Synonym	Klein four-ring
Related topic	Klein4Group
Related topic	InversesInRings
Related topic	NonCommutativeRingsOfOrderFour
Related topic	GroupsInField
Related topic	Subcommutative