Finding monic generator for ideals generated by polynomials of $\mathbb{Q}[X]$

Question

Let $R = \mathbb{Q}[X]$ be the ring of polynomials with rational coefficients and let:

$$I = \langle X^2 + 1 \rangle = \{(X^2 + 1)f(X) \ \vert \ f \in\mathbb{Q}[X] \} $$

$$J = \langle 2X + 1, X^2 - 4 \rangle = \{(2X + 1)f(X) + (X^2 - 4)g(X) \ \vert \ f,g \in \mathbb{Q}[X] \}$$

Find the monic generators for the ideals $I \cap J$ and $I + J$.

I think there's something wrong with this question. The ideal generated by $J$ is just all of $\mathbb{Q}[X]$, because:

$$1 = (2X + 1)\frac{(2X - 1)}{15} - \frac{4}{15}(X^2 - 4) \in J$$

so the monic generator for $I \cap J$ is just $X^2 + 1$, but then there's no monic generator for $I + J = \mathbb{Q}[X]$! Did I do something wrong here?

Answer 1

As you mentioned, $1\in J \iff J=\Bbb Q[X]$.

So, $I\cap J=I\cap \Bbb Q[X]=I=\langle X^2+1\rangle$ and $ I+J=I+\Bbb Q[X] =\Bbb Q[X]=\langle 1 \rangle.$

Also, note the general fact: $I=R\iff u\in I$, for some $u\in U(R)$, where $U(R)$ is the multiplicative group of the inverses, of the ring $R$.