Is there an algorithm to determine if we have been given a ring $A$ and its ideal $I$, whether or not $I$ is a maximal ideal of $A$? I found that sometimes proving that ideal is maximal might be tricky, like in here and a general algorithm is missing for example in Sage.
Let $R$ be the $\mathbb Q$ vector space generated by all ordinals of cardinality less than $c$ together with the ordinal $c$. Let multiplication be given by intersection. Let $I$ be the ideal generated by all countable ordinals. Since the continuum hypothesis is undecidable, it is undecidable whether $I$ is a maximal ideal.