example of Klein 4-ring

The Klein $4$ -ring $K$ can be represented by as a left ideal of $2\\times 2$ -matricesover the field with two elements $\\mathbb{Z}_{2}$ . Doing so helps to explainsome of the unnatural properties of this nonunital ring and is an example of howmany nonunital rings can often be understood as very natural subobjects of unitalrings.

K=\\left\\{\\begin{bmatrix}x&0\\\\y&0\\end{bmatrix}:x,y\\in\\mathbb{Z}_{2}\\right\\}

(1)

To match the product with with the table, use

	$\\displaystyle a$	$\\displaystyle:=\\begin{bmatrix}1&0\\\\0&0\\end{bmatrix},$
	$\\displaystyle b$	$\\displaystyle:=\\begin{bmatrix}0&0\\\\1&0\\end{bmatrix},$
	$\\displaystyle c$	$\\displaystyle:=\\begin{bmatrix}1&0\\\\1&0\\end{bmatrix}.$

Here the properties of the abstract multiplication table of $K$ (givenhere (http://planetmath.org/Klein4Ring)) can beseen as rather natural properties of unital rings. That is, the elements $a$ and $c$ are idempotents in the ring $M_{2}(\\mathbb{Z}_{2})$ and so theybehave similar to identities, and $b$ is nilpotent so that its annihilatingproperty is expected as well.

The second noncommutative nonunital ring of order 4 is the transpose of these matrices,that is, a right ideal of $M_{2}(\\mathbb{Z}_{2})$ .

Viewed in this way we recognize the Klein $4$ -ring as part of an infinite familyof similar nonunital rings of left/right ideals of a unital ring. Some authorsprefer to treat such objects only as ideals and not as rings so that the propertiesare always given the background of a more familiar structure.