Closed pairs

This is a study of closed pairs of abelian groups (closed elementary nets of degree 2). If the elementary group E (σ) does not contain new elementary transvections, then an elementary net σ (the net without the diagonal) is called closed. Closed pairs we construct from the subgroup of a polynomial ring. Let R1[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 R1[x]. Then the pair (A, B) is closed. In other words, if t12(β) or t21(α) is contained in subgroup 21(A), t12(B)>, then β\in B, α \in A.