Journal of the Ramanujan Mathematical Society

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On the Surjectivity of Certain Maps


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1 Stat Math Unit, Indian Statistical Institute, 8th Mile Mysore Road, RVCE Post, Bangalore - 560059, India
     

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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 ≥ 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.
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  • R. Ash, A course in algebraic number theory, Dover Publ., (2010) ISBN 978-0-486-47754-1.

  • M. F. Atiyah and I. G. Macdonald, Introduction to commutative algebra, Levant Books, Kolkata, First Indian Edition, (2007) ISBN 81-87169-87-7.

  • A. Frolich and M. J. Taylor, Algebraic number theory, Cambridge University Press, (1994) ISBN 0-521-43834-9.

  • R. W. Gilmer Jr., Integral domains which are almost Dedekind, Proc. Amer. Math. Soc., 15 (1964) 813–818. http://www.ams.org/journals/proc/1964-015-05/S0002-9939-1964-0166212-8/S0002-9939-1964-0166212-8.pdf.

  • S. Lang, Algebraic number theory, Springer Publ., (1986) ISBN 978-1-4612-6922-9.

  • K. A. Loper, Almost Dedekind domains which are not Dedekind, In: Brewer J. W., Glaz S., Heinzer W. J., Olberding B. M. (eds) Multiplicative Ideal Theory in Commutative Algebra, (2006), Springer, Boston, MA, ISBN 978-0-387-36717-0, 279–292, https://doi.org/10.1007/978-0-387-36717-0−17.

  • M. Nagata, A general theory of algebraic geometry over Dedekind domains I: The notion of models, American Journal of Mathematics, 78 no. 1, (1956) 78–116, https://www.jstor.org/stable/2372486.

  • M. Nagata, A general theory of algebraic geometry over Dedekind domains II: Separably generated extensions and regular local rings., American Journal of Mathematics, 80 no. 2, (1958) 382–420, https://www.jstor.org/stable/2372791.

  • M. Nagata, A general theory of algebraic geometry over Dedekind domains III: Absolutely irreducible models, simple spots, American Journal of Mathematics, 81 no. 2, (1959) 401–435, https://www.jstor.org/stable/2372749.

  • A. S. Rapinchuk, Strong approximation for algebraic groups, Thin Groups and Superstrong Approximation MSRI Publications, 61 (2013), http://library.msri.org/books/Book61/files/70rapi.pdf.

  • H. Uda, On a characterization of almost Dedekind domains, Hiroshima Math. J., 2 no. 2, (1972) 339–344, https://projecteuclid.org/euclid.hmj/1206137624.


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  • On the Surjectivity of Certain Maps

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Authors

C. P. Anil Kumar
Stat Math Unit, Indian Statistical Institute, 8th Mile Mysore Road, RVCE Post, Bangalore - 560059, India

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 ≥ 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.

References