PlanetMath: Klein 4-ring

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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

$\begin{displaymath}\begin{array}{c\vert cccc} \cdot & 0 & a & b & c \ \hline \... ...a \ \; b & 0 & b & 0 & b \ \; c & 0 & c & 0 & c \end{array}\end{displaymath}$

Note that this ring has two different right unities and .

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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Other names:

Klein's four-ring, Klein four-ring

Keywords:

non-commutative

This object's parent.

Attachments:: 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
There are 4 references to this entry.

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 3890 times total.

Classification:

AMS MSC:	16B99 (Associative rings and algebras :: General and miscellaneous :: Miscellaneous)
	20-00 (Group theory and generalizations :: General reference works )

Pending Errata and Addenda

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