Extended Hilbert’s Nullstellensatz

We prove the extended Hilbert’s Nullstellensatz in the context of Hu-Liu polynomial trirings. Which kinds of noncommutative rings are suitable for extending algebra geometry? Different attempts have been made to answer this question, but a satisfactory answer is still in hiding. My attempt at answering this question comes from the trivial extension of a ring by a bimodule over the ring. The trivial extension R ⊲< S of a ring R by a R-bimodule R S R has been used in both algebra geometry and commutative algebras for a long time ([1] and [5]). Even the R-bimodule R S R is itself a ring, the multiplicative structure on the ring R S R does not play any role in the trivial extension R ⊲< S, and the researchers who make use of the trivial extension R ⊲< S have not paid attension to the multiplicative structure on the ring R S R . Simply speaking, my idea of choosing a class of noncommutative rings is not to forget the multiplicative structure on the ring R S R while using the trivial extension R ⊲< S. If we combine the ring product on S with the bimodule actions on R S R by using the HuLiu triassociative law, then we get a triring structure on the trivial extension R ⊲< S, which was introduced in [2]. In particular, if R is a commutative ring and the R-bimodule R S R is also a commutative ring, then the resulting triring on R ⊲< S is called a Hu-Liu triring. A Hu-Liu triring R ⊲<S is a noncommutative ring with respect to the ring product on the trivial extension R ⊲< S if the left R-module R S is different from the right R-module S R . These kinds of Hu-Liu trirings are a class of noncommutative rings which are very close to commutative rings. Based on the curiosity to extend algebraic geometry in the context of HuLiu trirings, I started on the study of affine trialgebraic sets in Chapter 5 of [2]. This paper is the continuation of the study. The main result is the extended Hilbert’s Nullstellensatz.