# superRing -- Makes a super ring from two polynomial rings.

## Synopsis

• Usage:
R = superRing(R1, R2)
• Inputs:
• R1, ,
• R2, ,
• Outputs:
• R, , which has both invertible and skew symmetric variables, superRing

## Description

Let $R_1$ and $R_2$ be Two Polynomial rings on different set of variables A superRing is a new polynomial ring with three sets of variables. One set comes from $R_1$ and the second one is the inverse of it.

For example, if we have x as a variable in $R_1$, then there is a new variable, say $inverseVariable_0$ which is the inverse of $x$. The third set of variables comes from $R_2$. We redefine this set to be a set of skew-symmetric variables. So superRing of $R_1$ and $R_2$ is a quotient ring, which has both invertible and skew symmetric variables. If the coefficient ring is a field, then we get a super algebra.

 i1 : R1=QQ[x_1..x_5] o1 = R1 o1 : PolynomialRing i2 : R2=QQ[z_1..z_3] o2 = R2 o2 : PolynomialRing i3 : superRing(R1, R2) QQ[x ..x , inverseVariable ..inverseVariable , z ..z ] 1 5 0 4 1 3 o3 = ------------------------------------------------------------------------------------------------------------------------ (x inverseVariable - 1, x inverseVariable - 1, x inverseVariable - 1, x inverseVariable - 1, x inverseVariable - 1) 1 0 2 1 3 2 4 3 5 4 o3 : QuotientRing

## Ways to use superRing :

• "superRing(PolynomialRing,PolynomialRing)"

## For the programmer

The object superRing is .