Some basic question about a quotient ring

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Discussion Overview

The discussion revolves around the structure of the quotient ring (f1, f2)R/(x1, x2)R, where R is a complex polynomial ring in eight variables. Participants explore how the images of specific polynomials f1 and f2 appear in this quotient, examining the implications of the ideals involved.

Discussion Character

  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant questions the meaning of the notation (f1, f2)R/(x1, x2)R, seeking clarification on the numerator and denominator as ideals in R.
  • Another participant asserts that the numerator (f1, f2) is an ideal generated by f1 and f2, while the denominator (x1, x2) is an ideal generated by x1 and x2.
  • A participant suggests that in the quotient ring, all polynomial combinations of x1 and x2 are eliminated, leading to the proposal that \bar{f}1 = x5 x7 and \bar{f}2 = x6 x8.
  • Another participant agrees with the previous assertion, stating that the quotient ring cancels all terms containing x1 or x2.
  • A later reply posits that the quotient ring is generated by x5 x7 and x6 x8, proposing the equivalence (f1, f2)R/(x1, x2)R = (x5 x7, x6 x8).

Areas of Agreement / Disagreement

Participants express differing views on the exact nature of the images of f1 and f2 in the quotient ring, with some proposing specific forms while others seek clarification. The discussion remains unresolved regarding the complete characterization of the quotient ring.

Contextual Notes

There are unresolved assumptions about the nature of the ideals and the specific forms of the images of f1 and f2 in the quotient ring, which may depend on further clarification of the definitions involved.

naturemath
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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)?
 
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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)?
 
naturemath said:
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
 
> (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.
 
So what I'm interested in is: how do the images of f1 and f2 look like in (f1,f2)R/(x1,x2)R?
 
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 ?
 
yes. that quotient ring just cancels all terms containing x1 or x2.
 
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)
 

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