On the Surjectivity of Certain Maps

TL;DR

This paper establishes the surjectivity of three different algebraic maps related to ideals, projective spaces, and special linear groups, providing new insights into their algebraic properties and raising further questions.

Contribution

It proves the surjectivity of three specific maps in algebraic geometry and number theory, extending known results to new contexts and domains.

Findings

Surjectivity of the Chinese remainder reduction map for projective spaces.

Surjectivity of the strong approximation reduction map for rings with unital set condition.

Surjectivity of the map from k-dimensional special linear group to product of projective spaces.

Abstract

We prove in this article the surjectivity of three maps. We prove in Theorem $1.6$ the surjectivity of the Chinese remainder reduction map associated to the projective space of an ideal with a given factorization into ideals whose radicals are pairwise distinct maximal ideals. In Theorem $1.7$ we prove the surjectivity of the reduction map of the strong approximation type for a ring quotiented by an ideal which satisfies unital set condition. In Theorem $1.8$ we prove for Dedekind type domains which include Dedekind domains, for $k\geq 2$, the map from $k$-dimensional special linear group to the product of projective spaces of $k$ mutually co-maximal ideals associating the $k$-rows or $k$-columns is surjective. Finally this article leads to three interesting questions $[1.9, 1.10, 1.11]$ mentioned in the introduction section.