On the Surjectivity of Certain Maps II: For Generalized Projective Spaces

In this article we introduce generalized projective spaces (Definitions $[2.1, 2.5]$) and prove three main theorems in two different contexts. In the first context we prove, in main Theorem $A$, the surjectivity of the Chinese remainder reduction map associated to the generalized projective space of an ideal with a given factorization into mutually co-maximal ideals each of which is contained in finitely many maximal ideals, using the key concept of choice multiplier hypothesis (Definition $4.11$) which is satisfied. In the second context of surjectivity of the map from $k$-dimensional special linear group to the product of generalized projective spaces of $k$-mutually co-maximal ideals associating the $k$-rows or $k$-columns, we prove remaining two main Theorems $[\Omega,\Sigma]$ under certain conditions either on the ring or on the generalized projective spaces. Finally in the last section we pose open Questions $[9.1, 9.2]$ whose answers in a greater generality is not known.