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Subtraction of two non-homogenous monomial in a non-commutative ring

asked 2013-10-07 04:11:34 -0500

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Hi every one,

I have a problem for subtracting two non-homogenous monomial "xyyx-xyx" in a unital associative free algebra with two generators x&y.

The error which is appeared is "ArithmaticError : can only subtract the elements of the same degree".

I will apreciate some one who tell me what is the soloution.

Thanks

Abdolrasoul

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Actually the problem is here:

F.<x,y> = FreeAlgebra(QQ, implementation='letterplace') I=F[xyx-2xy]*F J=F.quo(I)

And the eroor is :

ArithmaticError : can only subtract the elements of the same degree"

Abdolrasoul Baharifard gravatar imageAbdolrasoul Baharifard ( 2013-10-07 09:26:39 -0500 )edit

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answered 2013-10-08 04:03:43 -0500

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The docstring of FreeAlgebra says

By http://trac.sagemath.org/7797, we provide a different implementation of free algebras, based on Singular's "letterplace rings". Our letterplace wrapper allows for chosing positive integral degree weights for the generators of the free algebra. However, only (weighted) homogenous elements are supported. Of course, isomorphic algebras in different implementations are not identical:

If you drop the implementation='letterplace' option, the ArithmeticError disappears:

sage: F.<x,y> = FreeAlgebra(QQ)
sage: x*y*x + 2*x*y
2*x*y + x*y*x
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But I need an unital associative free algebra and if I drop the "implementation= 'letterplace", it just give me free algebra.

Abdolrasoul Baharifard gravatar imageAbdolrasoul Baharifard ( 2013-10-08 08:39:01 -0500 )edit

Please use comments instead of "your answer". Maybe I am misunderstanding something, but how is the free algebra over `QQ` not unital?

Luca gravatar imageLuca ( 2013-10-08 10:03:33 -0500 )edit

Maybe I miss understood the definition of free algebra. When I put F.<x,y>=FreeAlgebra(QQ) "Sage" write me "Free Algebra en two generators x,y"" and when I write F.<x,y>=FreeAlgebra(QQ, implementation='letterplace') it gives me "Unital associative free algebra with two generators", but I think the both is the same, yes? Now, my problem is to construct a quotient of this algebra ba a non-homogenous ideal. Do you know, how can I do that?

Abdolrasoul Baharifard gravatar imageAbdolrasoul Baharifard ( 2013-10-09 06:27:15 -0500 )edit

See the example in the documentation by typing `F.quo?`

Luca gravatar imageLuca ( 2013-10-10 22:24:07 -0500 )edit

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Asked: 2013-10-07 04:11:34 -0500

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Last updated: Oct 08 '13