# Journal of the Ramanujan Mathematical Society

Volume 37, Issue 4, December 2022 pp. 347–373.

On the surjectivity of certain Maps III: The unital set condition

**Authors**:
C. P. Anil Kumar

**Author institution:**
Room. No. 223, Middle Floor, Main Building, Harish-Chandra Research Institute,
Chhatnag Road, Jhunsi, Prayagraj (Allahabad) 211 019, Uttar Pradesh, India.

**Summary: **
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 I =
∏{k}{i=1}I{k} with a given factorization into mutually co-maximal ideals I{j},1 ≤ j ≤
k where 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 Λ, the surjectivity of the
reduction map SP{2k}(R) → SP{2k} (R/I) of strong approximation type for a ring R
quotiented by an ideal I which satisfies the USC. In the third context, for a
positive integer k, we prove in the third main Theorem Ω, 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 Ij,0 ≤ j
≤ k associating the (k +1) -rows or (k +1)-columns, where the ideal I = ∏{k}
{j=0}I{j} satisfies the USC. In the fourth main Theorem Σ, 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 Ij,1 ≤ j ≤ 2k associating the (2k)-rows or (2k)-columns
where the ideal I = ∏{2k}{j=1}I{j} satisfies the USC. We also give counter
examples to analogous questions where surjectivity fails for (p,q)
-indefinite orthogonal groups over integers. Finally the answers to Questions
[1.1, 1.2, 1.3] in a greater generality are not known.

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