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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