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# Closed pairs

Koibaev, V. A.
Vladikavkaz Mathematical Journal 2011. Vol. 13. Issue 3.
Abstract:
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 $$\beta \in {B}$$, $$\alpha \in {A}$$.
Keywords: net, elementary net, closed net, net groups, elementary group, transvection.

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