Example of algebras over GF(2)

Lie

Anyone know of an example of an algebra over the field [tex]\mathbb{Z}_2[/tex] with the following properties?
1. commutative;
2. associative;
3. [tex] x^3 = 0 [/tex], for all x; and
4. Exists x and y such that [tex] x^2y \neq 0 [/tex].

Grateful!

micromass

What about [tex]\mathbb{Z}_2[X]/(X^3)[/tex]? It satisfies your first three properties, and alse the last one with y=1 and x=X...

Lie

micromass,

Note that condition 3 implies that the algebra can not have unity. Therefore [tex] \mathbb{Z}_2[X]/(X^3) [/tex] is not an example.

micromass

Oh sorry, I forgot to read "for all x" Well, I'll look for another example...

Hurkyl

What goes wrong with the direct way to approach the problem? (e.g. like micromass's, except working with algebras rather than rings)

Jamma

Ermm, can't you just take all polynomials over two variables x and y modulo the relation x^3=y^3=0 ?

Lie

Jamma, same remark:

Jamma

Sorry, I didn't mean it like that, I should describe my algebra a bit better.

Take as our set of elements {0,x,x^2,x^3,y,y^2,y^3} and all multiples and linear combinations of them with the obvious rules of addition and multiplication subject to the condition that x^3=y^3=0.

There is no unity here.

[Edit:ignore me, this algebra has elements in it which don't cube to zero]

Jamma

Ok, how about my algebra up there but with the relation (x)(y^2)=(x^2)(y).

It seems at a first glance that this works.