Hilbert’s Nullstellensatz forms the basis of algebraic geometry. In this paper we derive an analogous statement over the field of real numbers – the real Nullstellensatz. After that we introduce a generalization of real rings to the noncommutative setting. We prove a correspondence of ideals in the ring of real polynomials with the ideals in the ring of quaternionic polynomial functions. With this result we reduce the quaternionic Nullstellensatz to the real one. Finally, we state known generalizations and some further courses of study.
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