On the Surjectivity of Certain Maps III: For Symplectic Groups Over Rings and Generalized Projective Spaces

C. P. Anil Kumar

Published 2019-02-18Version 1

We prove three main theorems on the surjectivity of certain maps for symplectic groups over commutative rings with unity in two different contexts. In the first context, we prove in Theorem $\Lambda$, the surjectivity of the reduction map of strong approximation type for a ring quotiented by an ideal which satisfies unital set condition in the case of symplectic groups. In the second context of the surjectivity of the map from $(2k\times 2k)$-order symplectic group over a ring to the product of generalized projective spaces of $2k$-mutually co-maximal ideals associating the $2k$-rows or $2k$-columns, we prove the remaining two main Theorems $[\Omega,\Sigma]$, under certain conditions, either on the ring or on the generalized projective spaces. Finally in the second context, we give counter examples where, the surjectivity fails for $(p,q)$-indefinite orthogonal groups over integers.