Question: Let $R=\mathbb{Z}_{8}\oplus \mathbb{Z}_{30}$. Find all the maximal ideals of R and for each maximal ideal I, identify the size of the field R/I.
I am left stuck with this question.
Here are some helpful theorems:
Let I be a proper ideal of a commutative ring R. Then, I is a maximal ideal of R if B is an ideal of R and whenever $I\subseteq B\subseteq R$ then either I=B or I=R.
Let I be an ideal of a commutative ring R with unity and I is an ideal of R. Then, R/I is a field if and only if I is a maximal ideal of R.
Any help is appreciated. Please get me going. Hints preferably.
Thanks in advance.
Let $R$ be the ring $A \times B$.
Lemma: Every ideal of $R$ is of the form $I \times J$, where $I$ is an ideal of $A$, $J$ is an ideal of $B$.
(Hint: If $\mathfrak a$ is an ideal of $R$, you would obtain $I$ and $J$ by looking at the projection maps $R \rightarrow A, R \rightarrow B$, which are surjective ring homomorphisms)
Exercise: if $\mathfrak a$ is an ideal of $R$, give a nice description of $R/\mathfrak a$.
(Hint: using the lemma, it is exactly what you expect.)
Finally, note that if $S,T$ are nonzero rings, then the product $S \times T$ is never a field, even if $S$ and $T$ are.