Non-commutative algebra

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Concerning compositions of objects belonging to different classifications, the conventions adopted here are:. Set first an identifier for anticommutative and noncommutative variables using Setup , for example:. Consider now the list of objects. The first three are anticommutative, the next three are commutative and the last three are noncommutative:. See Also. Physics conventions.

Physics examples. Physics Updates. Tensors - a complete guide.

Noncommutative algebraic geometry

Mini-Course Computer Algebra for Physicists. So, to summarize the above. Commutative algebra will not only be more useful to you in number theory sooner, but will transfer more readily, and usefully, to other areas of mathematics you are likely to learn in the near future. I'll also point out that commutative algebra is at least half the foundation of modern algebraic geometry.

The objects of algebraic geometry are built by sticking together a collection of geometric structures associated to a commutative ring-so to study them, one must know plenty of commutative algebra. And if you're interested in going very far into number theory, you'll likely want to learn some algebraic geometry eventually. Sign up to join this community. The best answers are voted up and rise to the top. Home Questions Tags Users Unanswered.

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Ask Question. Asked 5 years, 10 months ago. Active 5 years, 10 months ago. Viewed 2k times. Sameer Kulkarni Sameer Kulkarni 8 8 silver badges 22 22 bronze badges.

I'm not sure about that, AlexYoucis : it might be non-commutative algebra includes group theory, without which almost nothing serious can be done in anything related to algebra somehow, up to and including number theory. In particular, usually the study of non-commutative algebras over a field. For example, one doesn't usually lump abelian group theory into commutative algebra.

Yet I agree in general it is meant to be non-comm.

Alex Youcis Alex Youcis Kevin Carlson Kevin Carlson 36k 3 3 gold badges 37 37 silver badges 78 78 bronze badges. In this case, the basis of D in degree 2 is.

Laudal : Noncommutative algebraic geometry

The output of normalElements tells us that in order for a degree 2 element of D to be normal, it must be an expresison in powers of the generators. The coefficients of these powers must satisfy the six equations listed. In Macaulay2, the user can define a polynomial ring to be commutative or skew-commutative the exterior algebra. Conversely, the user can convert an NCRing to a quotient of a polynomial ring in the usual sense of Macaulay2 using toM2Ring.

This method works on any NCRing - the result is the abelianization or "exterior-ization" of the given ring. For example, if we abelianize the skew polynomial ring C, we get a ring in which only powers of the generators are nonzero.

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On the other hand, if we "exterior-ize" C, we get the exterior algebra. Finally, we can construct the opposite ring.

The opposite ring of D will be the Ore extension by the inverse of sigma. See oppositeRing. Our last extended example illustrates how to obtain a presentation for the endomorphism ring of a module over a commutative ring. First we define a hypersurface ring and a high syzygy module. The endomorphism ring is computed using endomorphismRing.


This method computes a presentation, but the presentation is typically not minimal. To eliminate redundant generators and relations, use minimizeRelations. This method makes several passes through the presentation, and stops if no minimization occurs. We see a substantial reduction in the number of relations and that X 1 and X 3 are redundant generators.

The endomorphisms themselves are cached, and can be accessed via endomorphismRingGens. As an example, we explicitly verify that X 3 is redundant. Basic operations on noncommutative algebras The NCAlgebra package contains a number of methods for studying noncommutative rings - primarily graded rings.

Non-Commutative Algebras and Quantum Mechanics I with Basil Hiley