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A point model for the free cyclic submodules over ternions. (English) Zbl 1273.51004
For a commuative field $F$, the ring $T$ of upper triangular $2\times 2$ matrices over $F$ ist called the ring of ternions. It is shown that the set of all free cyclic submodules of $T^2$ admits a point model in $\mathrm{PG}(7,F)$ which is a smooth algebraic variety $\mathcal{X}\cup \mathcal{Y}$, where $\mathcal X$ corresponds to the unimodular submodules and $\mathcal Y$ (corresponding to the non-unimodular ones) is a line.
MSC:
51B99Nonlinear incidence geometry
51C99Ring geometry (Hjelmslev, Barbilian, etc.)
14J26Rational and ruled surfaces
16D40Free, projective, and flat modules and ideals (associative rings and algebras)
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References:
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