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If I and J are prime (radical) ideals then what are the conditions under which we can define a prime (radical) ideal from I+J?

Let $k$ be a field (not necessarily algebraically closed), define the $k$-algebra $$ B:=k[x_{0}, x_{1}, x_{2}, x_{3}]_{(F_{2}F_{3})} $$ (degree 0 part of the localization), it's the coordinate ring of ...