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We study systems of polynomial equations in several classes of finitelygenerated rings and algebras. For each ring $R$ (or algebra) in one of theseclasses we obtain an interpretation by systems of equations of a ring ofintegers $O$ of a finite field extension of either $\mathbb{Q}$ or$\mathbb{F}_p(t)$, for some prime $p$ and variable $t$. This implies that theDiophantine problem (decidability of systems of polynomial equations) in $O$ isKarp-reducible to the same problem in $R$. In several cases we further obtainan 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 thelong-standing conjecture that $\mathbb{Z}$ is always interpretable by systemsof equations in $O$ carries over to $R$. If true, it implies that theDiophantine 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 theunitary case was obtained by Eisentraeger); all possibly non-associative,non-commutative non-unitary rings that are f.g. as an abelian group; andseveral classes of f.g. non-commutative rings. Analogous statements areobtained for algebras over f.g. associative commutative unitary rings.  Another contribution is the technique by which the aforementioned results areobtained: 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 certainring (or algebra) $R$ with nice properties which is interpretable by systems ofequations in the multi-sorted structure $(A,B,C;f)$. This result is not onlyrelevant for rings and algebras, but also in other structures such as groups,as demonstrated previously by the authors.

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