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[parent] Klein 4-ring (Definition)

One of the two smallest non-commutative ring 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 $\{0,\,b\}$




"Klein 4-ring" is owned by pahio.
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See Also: Klein 4-group, inverses in rings, non-commutative rings of order four, groups in field

Other names:  Klein's four-ring, Klein four-ring
Keywords:  non-commutative

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example of Klein 4-ring (Example) by Algeboy
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Cross-references: two-sided ideal, subrings, right unities, binary operation, neutral element, Klein 4-group, ring, non-commutative
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This is version 11 of Klein 4-ring, born on 2005-02-06, modified 2008-03-11.
Object id is 6720, canonical name is Klein4Ring.
Accessed 4897 times total.

Classification:
AMS MSC16B99 (Associative rings and algebras :: General and miscellaneous :: Miscellaneous)
 20-00 (Group theory and generalizations :: General reference works )

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