# Quotients of exterior algebras

I am trying to take a quotient of an exterior algebra by a two-sided ideal, and having some trouble. As an easy case to understand the syntax, I tried to compute $\Lambda^\mathbb{Q}[x,y,z]/(x-y)$, where I would expect $\overline{x} = \overline{y}$. However, it seems to not consider xbar and ybar the same.

sage: E.<x,y,z> = algebras.Exterior(QQ);
sage: I = E.ideal(x-y);
sage: Q = E.quo(I);
sage: xbar,ybar,zbar = Q.gens();
sage: xbar == ybar
False


I thought maybe that was fine because xbar and ybar are generators of the quotient so maybe there is some funkiness going on there. So then I tried to take the image of x and y under the quotient map and check if they are equal, but it still said they were not.

sage: q = Q.cover()
sage: q(x) == q(y)
False


Am I misunderstanding something in Sage? In the algebra of it? Both?

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Ok it looks like this is the current (Sage 9.3beta8) behavior.

In the source code for Ideal_nc, which is the type of I, there is no implementation of a .reduce() method. Since Ideal_nc is a subclass of Ideal_generic it defaults to the Ideal_generic.reduce() method. But when Ideal_generic.reduce() is called it defaults to the identity:

    def reduce(self, f):
r"""
Return the reduction of the element of f modulo self.

This is an element of R that is equivalent modulo I to f where
I is self.

EXAMPLES::

sage: ZZ.ideal(5).reduce(17)
2
sage: parent(ZZ.ideal(5).reduce(17))
Integer Ring
"""
return f       # default


So this is asking if xbar and ybar are literally the same, which they are not.

Thanks to Travis Scrimshaw for pointing this out to me in private communication.

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