= Solution
The <first isomorphism theorem for rings> states that a ring homomorphism $\phi:R\to S$ induces an isomorphism
$$
R/\ker\phi\cong\operatorname{im}\phi.
$$
Because $J$ is an <ideal>, sums and products of elements of $R+J$ remain in $R+J$, so it is a subring. The surjective homomorphism
$$
R\longrightarrow(R+J)/J,
\qquad r\longmapsto r+J
$$
has kernel $R\cap J$. The theorem gives
$$
\boxed{R/(R\cap J)\cong(R+J)/J}.
$$
Evaluation at $-2$ gives
$$
\mathbb Q[X]/(X+2)\cong\mathbb Q,
$$
so this quotient is a <field> of <characteristic> zero. The other quotient is
$$
\mathbb Z[X]/(3,X^2+X+1)
\cong\mathbb F_3[X]/((X-1)^2).
$$
It has characteristic $3$ but is not a field, since the nonzero class of $X-1$ is <nilpotent element>[nilpotent].
Back to article page
