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"Ratio" of two elements in a ring

Suppose I have a 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 Q-span of y with 1-dimensional vector space, choose an isomorphism of this vector space with Q (sending y to 1), and see where x goes.

The general context is that R is a finite-dimensional graded 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 Q (defined by sending a certain element to 1), and I want to be able to compute what it does to other elements.

"Ratio" of two elements in a ring

Suppose I have a 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 Q-span of y y with 1-dimensional vector space, choose an isomorphism of this vector space with Q (sending y to 1), and see where x goes.

The general context is that R is a finite-dimensional graded 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 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=Q[x1,x2,x3,x4]. Let I1=(x1x3,x2x4). Let I2=(x1+x3,x2+x4). Let R=A/(I1+I2).

(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=x1x2, and let x=(x1+x3)(x2+x4). 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 = 4y.)

"Ratio" of two elements in a ring

Suppose I have a 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 Q-span of y with 1-dimensional vector space, choose an isomorphism of this vector space with Q (sending y to 1), and see where x goes.

The general context is that R is a finite-dimensional graded 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 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=Q[x1,x2,x3,x4]. Let I1=(x1x3,x2x4). Let I2=(x1+x3,x2+x4). Let R=A/(I1+I2).

(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=x1x2, and let $x = (x_1 + x_3)(x_2 + x_4)$. 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 = 4y.)

"Ratio" of two elements in a ring

Suppose I have a 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 Q-span of y with 1-dimensional vector space, choose an isomorphism of this vector space with Q (sending y to 1), and see where x goes.

The general context is that R is a finite-dimensional graded 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 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=Q[x1,x2,x3,x4]. Let I1=(x1x3,x2x4). Let I2=(x1+x3,x2+x4). Let R=A/(I1+I2).

(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=x1x2, and let x=(x1+x4)(x2+x3). 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 = 4y.)2y.)

Later edit: fixed typo