# Journal of the Ramanujan Mathematical Society

Volume 35, Issue 3, September 2020 pp. 241–262.

On the surjectivity of certain maps II: for generalized projective spaces

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

**Author institution:**Flat No. 104, Bldg. No. 23, Lakshmi Paradise, 5th Main, 11th Cross, Lakshmi Narayana Puram, Near Muneeswaraswamy Narasimhaswamy Temple, Bengaluru 560 021, Karnataka, India

**Summary: **
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 only a 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 [Ω, Σ]
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 are not known.

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