# Determining if any of these three are an ideal of $\mathbb{R}[x]$

$$\mathbb{R}[x]$$ denotes the ring of polynomials in $$x$$ with real coefficients. Let $$I \subset \mathbb{R}[x]$$ be the subset of those polynomials with constant coefficient $$0$$, and let $$J \subset \mathbb{R}[x]$$ be the set of polynomials with linear coefficient $$0$$: $$I = \{a_1x+a_2x^2+\dots+a_nx^n|n\geq 0,a_i\in\mathbb{R}\}$$ $$J = \{b_0+b_2x^2+\dots+b_mx^m|m\geq 0,b_i\in\mathbb{R}\}$$ i) Is $$I$$ an ideal of $$\mathbb{R}[x]$$?

ii) Is $$J$$ an ideal of $$\mathbb{R}[x]$$?

iii) Is $$I\cap J$$ an ideal of $$\mathbb{R}[x]$$?

i)$$I = \mathbb{R}[x] / a_0$$ so we need only compare $$a_0 * I$$ and $$I * a_0$$

$$a_0 * I = {a_0(a_1x+a_2x^2+\dots+a_nx^n)\in I}$$ $$I * a_0= {(a_1x+a_2x^2+\dots+a_nx^n)a_0\in I}$$ Hence $$I$$ is an ideal of $$\mathbb{R}[x]$$

ii) Proof $$J$$ is not an ideal by counterexample:

$$xJ = {xb_0 + b_2x^3 + b_3x^4+\dots+b_mx^m+1}\not\in J$$

iii) $$I \cap J$$ in an ideal of $$\mathbb{R}[x]$$

$$I \cap J = \{b_2x^2+\dots+b_mx^m|m\geq 0,b_i\in\mathbb{R}\}$$

Laptop is lagging to a halt now, but it is an ideal, by same logic as i)

Are these correct?

1. $I$ is the ideal generated by the polynomial $x$.
2. $J$ is not an ideal because $1\in J$ and $x\notin J$.
3. $I\cap J$ is the ideal generated by the polynomial $x^2$.
• Can you explain $2$? Jun 24 '14 at 9:10
• @Examin5days You should explain it! ;-) It is the required counterexample, isn't it? Since $1\in J$, for every polynomial $f$ you should have $1\cdot f\in J$. Jun 24 '14 at 9:11