## Example of algebras over GF(2)

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 $$x^2y \neq 0$$.

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 Blog Entries: 8 Recognitions: Gold Member Science Advisor Staff Emeritus 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...
 micromass, Note that condition 3 implies that the algebra can not have unity. Therefore $$\mathbb{Z}_2[X]/(X^3)$$ is not an example.

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## Example of algebras over GF(2)

Oh sorry, I forgot to read "for all x" Well, I'll look for another example...
 Recognitions: Gold Member Science Advisor Staff Emeritus What goes wrong with the direct way to approach the problem? (e.g. like micromass's, except working with algebras rather than rings)
 Ermm, can't you just take all polynomials over two variables x and y modulo the relation x^3=y^3=0 ?

 Quote by Jamma 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:
 Quote by Lie micromass, Note that condition 3 implies that the algebra can not have unity.[...]
 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]
 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.

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