An abelian ambient category for behaviors in algebraic systems theory

We describe an abelian category $\mathbf{ab}(M)$ in which the solution sets of finitely many linear equations over an arbitrary ring $R$ with values in an arbitrary left $R$-module $M$ reside as objects. Such solution sets are also called behaviors in algebraic systems theory. We both characterize $\mathbf{ab}(M)$ by a universal property and give a construction of $\mathbf{ab}(M)$ as a Serre quotient of the free abelian category generated by $R$. We discuss features of $\mathbf{ab}(M)$ relevant in the context of algebraic systems theory: if $R$ is left coherent and $M$ is an fp-injective fp-cogenerator, then $\mathbf{ab}(M)$ is antiequivalent to the category of finitely presented left $R$-modules. This provides an alternative point of view to the important module-behavior duality in algebraic systems theory. We also obtain a dual statement: if $R$ is right coherent and $M$ is fp-faithfully flat, then $\mathbf{ab}(M)$ is equivalent to the category of finitely presented right $R$-modules. As an example application, we discuss delay-differential systems with constant coefficients and a polynomial signal space. Moreover, we propose definitions of controllability and observability in our setup.