example of Klein 4-ring
The Klein -ring can be represented by as a left ideal of -matrices over the field with two elements . Doing so helps to explain some of the unnatural properties of this nonunital ring and is an example of how many nonunital rings can often be understood as very natural subobjects of unital rings.
To match the product with with the table, use
Here the properties of the abstract multiplication table of (given here (http://planetmath.org/Klein4Ring)) can be seen as rather natural properties of unital rings. That is, the elements and are idempotents in the ring and so they behave similar to identities, and is nilpotent so that its annihilating property is expected as well.
Viewed in this way we recognize the Klein -ring as part of an infinite family of similar nonunital rings of left/right ideals of a unital ring. Some authors prefer to treat such objects only as ideals and not as rings so that the properties are always given the background of a more familiar structure.
|Title||example of Klein 4-ring|
|Date of creation||2013-03-22 17:41:59|
|Last modified on||2013-03-22 17:41:59|
|Last modified by||Algeboy (12884)|