Studying the Diophantine problem in finitely generated rings and algebras via bilinear maps

We study systems of polynomial equations in several classes of finitely generated rings and algebras. For each ring $R$ (or algebra) in one of these classes we obtain an interpretation by systems of equations of a ring of integers $O$ of a finite field extension of either $\mathbb{Q}$ or $\mathbb{F}_p(t)$, for some prime $p$ and variable $t$. This implies that the Diophantine problem (decidability of systems of polynomial equations) in $O$ is Karp-reducible to the same problem in $R$. In several cases we further obtain an interpretation by systems of equations of the ring $\mathbb{F}_p[t]$ in $R$, which implies that the Diophantine problem in $R$ is undecidable in this case. Otherwise, the ring $O$ is a ring of algebraic integers, and then the long-standing conjecture that $\mathbb{Z}$ is always interpretable by systems of equations in $O$ carries over to $R$. If true, it implies that the Diophantine problem in $R$ is also undecidable. Some of the classes of f.g. rings studied in this paper are the following: all associative, commutative, non-unitary rings (a similar statement for the unitary case was obtained by Eisentraeger); all possibly non-associative, non-commutative non-unitary rings that are f.g. as an abelian group; and several classes of f.g. non-commutative rings. Analogous statements are obtained for algebras over f.g. associative commutative unitary rings. Another contribution is the technique by which the aforementioned results are obtained: We show that given a bilinear map $f: A\times B \to C$ between f.g. abelian groups (or modules), under mild assumptions, there exists a certain ring (or algebra) $R$ with nice properties which is interpretable by systems of equations in the multi-sorted structure $(A,B,C;f)$. This result is not only relevant for rings and algebras, but also in other structures such as groups, as demonstrated previously by the authors.