Does Anyone Know an Example of an Algebra Over GF(2) With Specific Properties?

[Lie]

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

Grateful!

 

[micromass]

What about \mathbb{Z}_2[X]/(X^3)? 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 \mathbb{Z}_2[X]/(X^3) 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]

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

Jamma, same remark:

micromass,

Note that condition 3 implies that the algebra can not have unity.[...]

 

[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.