Since this is somewhat hidden in the comments, let me give the following answer:

• The statement that the sum of two nilpotents is nilpotent is so basic that it seems to be used in the construction and the verification of the Zariski locale/topos/lattice. I don't think that constructive algebra can prove this without circular arguments.
• However, the statement $I+J=A \Rightarrow I^n+J^m=A$ can be proven by working in the lattice of radical ideals: $$\sqrt{I^n+J^m}=\sqrt{I^n} \vee \sqrt{J^m}=\sqrt{I} \vee \sqrt{J}=\sqrt{I+J}=A.$$ Whenever one uses the open subset $D(I)$ in a proof, one may simply replace it by the radical ideal $\sqrt{I}$