Closed pairs

This is a study of closed pairs of abelian groups (closed elementary nets of degree 2). If the elementary group $E(\sigma)$ does not contain new elementary transvections, then an elementary net $\sigma$ (the net without the diagonal) is called closed. Closed pairs we construct from the subgroup of a polynomial ring. Let $R_1[x]$ – the ring of polynomials (of variable $x$ with coefficients in a domain $R$) with zero constant term. We prove the following result.
Theorem. Let $A,B$ – additive subgroups of $R_1[x]$. Then the pair $(A,B)$ is closed. In other words, if $t_{12}(\beta)$ or $t_{21}(\alpha)$ is contained in subgroup $\langle t_{21}(A),t_{12}(B)\rangle$, then $\beta\in B$, $\alpha\in A$.