Compute the elements of an Ideal of a polynomial ring

abstract-algebragroup-theoryidealsring-theory

I am reading about polynomial quotient rings and the text starts with

Let $\Bbb R[x]$ denote the ring of polynomials with real coefficients and let $\langle x^2 + 1\rangle $ denote the principal ideal generated by $x^2 + 1$; that is,

$$\Bbb R[x]/ \langle x^2 + 1 \rangle= \{f(x)(x^2 + 1)\mid f(x) \in\Bbb R[x]\}.$$

Even before I try to understand the cosets & the quotient group, I am having trouble understanding how the Ideal $\langle x^2 + 1 \rangle$ is generated & what are the elements of the Ideal.

So I wanted to try it out with a smaller group instead of the infinite group of reals $\Bbb R$.

Can I generate the ideal for something thing like $\Bbb Z_5[x]$ or $\Bbb Z_3[x]$? What would be the elements of the Ideal $\langle x^2 + 1 \rangle$ of a ring $\Bbb Z_5[x]$ or $\Bbb Z_3[x]$?

I would have to compute all $f(x)(x^2+1)$ for all $f(x) \in \Bbb Z_3(x)$

So would it be first iterating all the element of $\Bbb Z_3(x)$

i.e.

$0, 1, 2, x, x + 1, x + 2, 2x, 2x+1, 2x+2$ and so on and then adding these elements to $x^2+1$?

Best Answer

Assuming you mostly care about commutative rings (so we don't have to talk about left vs right ideals), the elements in an ideal are just all linear combinations of products of elements of the ring with the generators. This always includes, for example, integer multiples of the generators. (It makes sense to say "integer multiple" regardless of the ring, as long as we understand that we might have $3=0$, for example.)

So in the ring $\Bbb Z[x]$, the ideal generated by $x^2+1$ contains

  • $0$
  • $x^2+1$, $-(x^2+1)$, $2(x^2+1)$, $-2(x^2+1)$, etc,
  • $x(x^2+1)$, $-x(x^2+1)$, $2x(x^2+1)$, $-2x(x^2+1)$, etc,
  • $x^2(x^2+1)$, $-x^2(x^2+1)$, $2x^2(x^2+1)$, $-2x^2(x^2+1)$, etc,
  • etc.

When $R$ is infinite, ideals often contain infinitely many elements. A useful way to think about ideals is as an additive subgroup which is closed under multiplication by elements of your ring.

All of the elements of the ideal generated by $x^2+1$ in $\Bbb Z_3[x]$ would be

  • $0$
  • $x^2+1$ and $2x^2+2$
  • $x^3+x$ and $2x^3+2x$
  • $x^4+x^2$ and $2x^4+2x^2$
  • etc.

Note that $\Bbb Z_3[x]$ is still infinite, so the ideal has infinitely many elements. A simpler example to think about is the ideal generated by $2$ in $\Bbb Z$. This consists of exactly the even integers $0,\pm 2,\pm 4,\ldots$. (The odd integers don't form an ideal because $2(1)=2$ is not odd.)

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