I'm trying to find an algebraic set $V$ that can be written as the disjoint union of two proper algebraic sets, such that the coordinate ring $k[V]$, where $k$ is NOT algebraically closed, is NOT the direct sum of two ideals. I proved that if $k$ is algebraically closed, then the coordinate ring must in fact decompose into a direct sum, but I'm having trouble finding a counterexample when the field is not algebraically closed. I tried $\mathbb R$ and $x^2+1$, and I tried a finite field, but I couldn't get either to work.
Could I just have a hint as to what type of counterexample I should be looking for?
If your definition of intersect is that $I + J$ is proper (i.e. there is a maximal ideal containing both), then it's not possible to do this for a reduced f.g. $k$-algebra $R$: Assume we have $R$ and $I,J$ ideals such that $I+J = R$ and such that every maximal ideal contains $IJ$. In a reduced f.g. $k$-algebra, the intersection of all maximal ideals is $0$, so $IJ = 0$ (rings that satisfy this are called Jacobson)$.
Then there exists $i \in I$ such that $i$ is congruent to $1 \ mod \ J$ and $j \in J$ such that $J$ is congruent to $1$ mod $I$.
Then take the map $R/I \times R/J \to R$ given by $r+I \times s+J \mapsto rj+si$. Here we use that $IJ = 0$ to show that it is well defined. The inverse map is given by $r \mapsto r+I \times r+J$.