Some basic question about a quotient ring

[naturemath]

It's been awhile since I studied ring theory but here's a question about it:

Let R = C[x₁, x₂, x₃, x₄, x₅, x₆, x₇, x₈] be a complex polynomial ring in 8 variables.

Let

f₁ = x₁ x₃ +x₅ x₇ and
f₂ = x₂ x₄ +x₆ x₈.

How do \bar{f}₁, \bar{f}₂ in (f₁,f₂)R/(x₁,x₂)R look like?


Is it
\bar{f}₁ = x₅ x₇ + (x₁,x₂)

\bar{f}₂ = x₆ x₈ + (x₁,x₂)

or is it

\bar{f}₁ = x₁ x₃ + x₅ x₇ +(x₁,x₂)

\bar{f}₂ = x₂ x₄ + x₆ x₈ + (x₁,x₂)?

 

[naturemath]

Since

x₁*x₃ \in (x₁, x₂),

x₁ x₃+ x₅ x₇ ~ x₁ x₃ mod (x₁, x₂).

So the coset x₁ x₃+ x₅ x₇ + (x₁, x₂) = x₅ x₇ + (x₁, x₂)?

 

[DonAntonio]

It's been awhile since I studied ring theory but here's a question about it:

Let R = C[x₁, x₂, x₃, x₄, x₅, x₆, x₇, x₈] be a complex polynomial ring in 8 variables.

Let

f₁ = x₁ x₃ +x₅ x₇ and
f₂ = x₂ x₄ +x₆ x₈.

How do \bar{f}₁, \bar{f}₂ in (f₁,f₂)R/(x₁,x₂)R look like?

What is that?? What do you mean by \,\,(f_1,f_2)R/(x_1,x_2)R\,\,? The denominator clearly is

an ideal, but what is the numerator?

Is it
\bar{f}₁ = x₅ x₇ + (x₁,x₂)

\bar{f}₂ = x₆ x₈ + (x₁,x₂)

or is it

\bar{f}₁ = x₁ x₃ + x₅ x₇ +(x₁,x₂)

\bar{f}₂ = x₂ x₄ + x₆ x₈ + (x₁,x₂)?

DonAntonio

 

[naturemath]

> (f1,f2)R/(x1,x2)R?

The numerator (f1, f2) is an ideal in R generated by f1 and f2 while the denominator (x1, x2) is an ideal in R generated by x1 and x2.

 

[naturemath]

So what I'm interested in is: how do the images of f1 and f2 look like in (f1,f2)R/(x1,x2)R?

 

[naturemath]

It seems to me that in (f1,f2)R/(x1,x2)R, we are killing off all polynomial combination of x1 and x2 (all polys of the form ax1 + bx2 where a and b are in R), or are we killing off something smaller than that?


Thus, do

\bar{f}₁ = x₅ x₇


\bar{f}₂ = x₆ x₈

in (f1,f2)R/(x1,x2)R ?

 

[mathwonk]

yes. that quotient ring just cancels all terms containing x1 or x2.

 

[naturemath]

So (f1,f2)R/(x1,x2)R is generated by x5 x7 and x6 x8, right?

I.e., (f1,f2)R/(x1,x2)R = (x5 x7, x6 x8)