A noncommutative ring is a Ring of subclass FreeAlgebra or FreeAlgebraQuotient.
In addition to defining a ring as a quotient of a FreeAlgebra, some common ways to create noncommutative rings include skewPolynomialRing and oreExtension.
Let's consider a three dimensional Sklyanin algebra. We first define the free algebra on the variables x,y,z:
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Then input the defining relations, and put them in an ideal:
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Next, we will define the quotient ring (as well as try a few functions on this new ring). Note that when the quotient ring is defined, Macaulay2 computes the Groebner basis of I (out to a certain degree, should the Groebner basis be infinite).
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As we can see, $x$ is now an element of the quotient $B$.
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If we define a new ring containing x, x is now part of that new ring. For example, we can use the following command to define the (-1)-skew polynomial ring on the variables x,y,z,w:
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We can 'go back' to B using the command use(Ring).
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Then call the command oreExtension.
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