On the Surjectivity of Certain Maps III: The Unital Set Condition

In this article, for generalized projective spaces with any weights, we prove four main theorems in three different contexts where the Unital Set Condition USC (Definition $2.8$) on ideals is further examined. In the first context we prove, in the first main Theorem $A$, the surjectivity of the Chinese remainder reduction map associated to the generalized projective space of an ideal $\mathcal{I}=\underset{i=1}{\overset{k}{\prod}}\mathcal{I}_k$ with a given factorization into mutually co-maximal ideals $\mathcal{I}_j,1\leq j\leq k$ where $\mathcal{I}$ satisfies the USC, using the key concept of choice multiplier hypothesis (Definition $4.10$) which is satisfied. In the second context, for a positive $k$, we prove in the second main Theorem $\Lambda$, the surjectivity of the reduction map $SP_{2k}(\mathcal{R})\rightarrow SP_{2k}(\frac{\mathcal{R}}{\mathcal{I}})$ of strong approximation type for a ring $\mathcal{R}$ quotiented by an ideal $\mathcal{I}$ which satisfies the USC. In the third context, for a positive integer $k$, we prove in the thrid main Theorem $\Omega$, the surjectivity of the map from special linear group of degree $(k+1)$ to the product of generalized projective spaces of $(k+1)$-mutually co-maximal ideals $\mathcal{I}_j,0\leq j\leq k$ associating the $(k+1)$-rows or $(k+1)$-columns, where the ideal $\mathcal{I}=\underset{j=0}{\overset{k}{\prod}}\mathcal{I}_j$ satisfies the USC. In the fourth main Theorem $\Sigma$, for a positive integer $k$, we prove the surjectivity of the map from the symplectic group of degree $2k$ to the product of generalized projective spaces of $(2k)$-mutually co-maximal ideals $\mathcal{I}_j,1\leq j\leq 2k$ associating the $(2k)$-rows or $(2k)$-columns where the ideal $\mathcal{I}=\underset{j=1}{\overset{2k}{\prod}}\mathcal{I}_j$ satisfies the USC. The answers to Questions [1.1,1.2,1.3] in a greater generality are not known.