# Thread: Several problems regarding Ideals

1. ## Several problems regarding Ideals

Hello, i am stumped on these problems, and i am wondering if anybody could help me

1. Prove or disprove that in the ring Z[X], <2X> = <2,X>

2. Suppose that I is an ideal of Q[X] which contains both X^{2} + 2X + 4 and X^{3} - 3. Show that I = Q[X].

3. Prove that in the ring Z[X], <2>U<X> is not an ideal

2. Originally Posted by maroon_tiger
1. Prove or disprove that in the ring Z[X], <2X> = <2,X>
To show that $\left< 2x \right> = \left< 2,x \right>$ we need to show $\left< 2x \right> \subseteq \left<2 ,x\right>$ and $\left<2,x\right> \subseteq \left<2x\right>$.

2. Suppose that I is an ideal of Q[X] which contains both X^{2} + 2X + 4 and X^{3} - 3. Show that I = Q[X].
Note $x^3 - 3 = (x-2)(x^2+2x+4) + 5$ this means $\gcd (x^3 - 3,x^2+2x+4) = \gcd (x^2+2x+4, 5) = 1$ by Euclidean algorithm. Thus, by relative primeness there exists $f(x),g(x)\in \mathbb{Q}[x]$ such that $f(x)(x^2+2x+4) + g(x) (x^3-3) = 1$. Since $I$ is an ideal it means $f(x)(x^2+2x+4) + g(x) (x^3-3) = 1 \in I$. All ideals which contain $1$ have to be the improper ideal, i.e. $I = \mathbb{Q}$.

3. Prove that in the ring Z[X], <2>U<X> is not an ideal
Find two elements $a,b\in \left< 2\right> \cup \left< x \right>$ so that $a+b\not \in \left<2\right> \cup \left< x \right>$.

3. thanks for the help

4. Originally Posted by ThePerfectHacker
To show that $\left< 2x \right> = \left< 2,x \right>$ we need to show $\left< 2x \right> \subseteq \left<2 ,x\right>$ and $\left<2,x\right> \subseteq \left<2x\right>$.
i tried that but i believe that they arent equal b/c x^{3} + 2X^{2} + 2 has a factor of <2,X> b/c of 2(X^{2} +1) + X(X^{2}) but x^{3} + 2X^{2} + 2 doesnt have a factor of <2x> unless im wrong

also, on another problem,

1. Prove that in the ring Z[X], <2> /\ <X> = <2x>
i understand why this would be true. would i prove this by the properties of an ideal? by assuming that <2> = {2f(x), x in some ideal A} and <X> = {Xg(y). some y in B}, then a point "p" in <2>^<X> such that {p exists in 2f(x)Xg(x)} and prove that 2f(x)Xg(x) = 2xf(x)? is that correct?