# "Ratio" of two elements in a ring

Suppose I have a $\mathbb{Q}$-algebra R, and I have two elements x and y in R. I happen to know that x is a scalar multiple of y. Is there a way to figure out what the scalar is?

In other words, I want to identify the $\mathbb{Q}$-span of y with 1-dimensional vector space, choose an isomorphism of this vector space with $\mathbb{Q}$ (sending y to 1), and see where x goes.

The general context is that R is a finite-dimensional graded $\mathbb{Q}$-algebra (given as a quotient of a polynomial ring), and the top degree piece has dimension 1. I have an isomorphism of this top degree piece with $\mathbb{Q}$ (defined by sending a certain element to 1), and I want to be able to compute what it does to other elements.

An example of such a ring: let $A = \mathbb{Q}[x_1, x_2, x_3, x_4]$. Let $I_1 = (x_1x_3, x_2x_4)$. Let $I_2 = (x_1 + x_3, x_2 + x_4)$. Let $R = A/(I_1 + I_2)$.

(Note that $R$ is the Stanley-Reisner ring of simplicial complex (a triangulation of the circle) modulo a linear system of parameters. So it is graded by degree, and as the simplicial complex is a circle, the degree 2 is a 1-dimensional vector space.)

Let $y = x_1 x_2$, and let $x = (x_1 + x_4)(x_2 + x_3)$. Then x is a scalar multiple of y (as they are both in degree 2), and I would like to know what the scalar is.

(In this case, x = 2y.)

Later edit: fixed typo

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You can use the normal basis associated to a Gröbner basis. Please add an example of R.

( 2021-04-08 08:32:15 +0200 )edit

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This is quite straightforward:

A.<x1,x2,x3,x4> = PolynomialRing(QQ)
I12 = ideal(x1*x3,x2*x4) + ideal(x1+x3,x2+x4)
R = A.quotient_ring(I12)
print( R((x1+x4)*(x2+x3)) / R(x1*x2) )

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