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Example of algebras over GF(2)

by Lie
Tags: algebra, algebra help
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Lie
#1
May8-11, 08:52 AM
P: 15
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].

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micromass
#2
May8-11, 09:21 AM
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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
#3
May8-11, 10:55 AM
P: 15
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
#4
May8-11, 11:10 AM
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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...
Hurkyl
#5
May8-11, 04:00 PM
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What goes wrong with the direct way to approach the problem? (e.g. like micromass's, except working with algebras rather than rings)
Jamma
#6
May8-11, 06:01 PM
P: 429
Ermm, can't you just take all polynomials over two variables x and y modulo the relation x^3=y^3=0 ?
Lie
#7
May9-11, 12:07 PM
P: 15
Quote Quote by Jamma View Post
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 Quote by Lie View Post
micromass,

Note that condition 3 implies that the algebra can not have unity.[...]
Jamma
#8
May9-11, 02:12 PM
P: 429
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
#9
May9-11, 02:22 PM
P: 429
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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