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  <title>Research</title>
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  <h1>Research</h1>
  <p align="justify"> My broad research area is Commutative Algebra. In commutative algebra,
 we study commutative rings, ideals, and modules over commutative rings. The concept of a module
 over a ring is a generalization of the concept of vector space over a field. During my Ph.D, 
I studied projective modules, unimodular rows and its topological analogue. I am also 
interested to work in the area of Algebraic Geometry. Currently, I am working on 
multiplicative Lie algebra, Ore extension of a ring, and algebraic analog of the cohomotopy 
group.<br/></p>

<h2>Publications</h2>
<ul class="custom"><li align="justify"> <b>Existence of a unique group of finite order</b>, The Mathematics Student, 81 
(1 - 4 ), 215 - 218, 2012 (with Prof. Shiv Datt Kumar). </li><br>
<li align="justify"><b>On the number of generators of a projective module</b>, Proc. Indian Acad. Sci. (Math. Sci.), 123 (4), 469 - 478, November 2013
  (with Prof. Shiv Datt Kumar and Prof. Raja Sridharan).</li><br>
<li align="justify"><b>On completable unimodular row over a polynomial ring</b>, J. Indian Math. Soc. 82 (1 - 2),  185 - 195, January - June 2015.</li><br>
<li align="justify"><b>Connection between commutative algebra and topology</b>, Indian J. Pure Appl. Math., 47 (3), 471-489,
 September 2016 (with Prof. Shiv Datt Kumar and Prof. Raja Sridharan).
</li><br>
<li align="justify"><b>Projective modules and efficient generation of ideals over Laurent polynomial rings</b>, Proc. Natl. Acad. Sci., India, Sect. A Phys. Sci. , 
87(3), 349-353, September 2017 (with Prof. Shiv Datt Kumar and Prof. Raja Sridharan).</li><br>
<li align="justify"><b>Multiplicative Lie algebras and Schur multiplier</b>, J. Pure Appl. Algebra, 223 (9), 
3695-3721, September 2019, (with Prof. Ramji Lal).</li><br>
<li align="justify"><b>A construction of Bent functions on a finite group</b>,  Nat. Acad. Sci. Lett., 43 (3), 247-251, June 2020 (with Mani Shankar Pandey and Dr. Vipul Kakkar).</li><br>
<li align="justify">The Chapter "<b>On an algebraic analogue of the Mayer-Vietoris sequence</b>", Leavitt 
Path Algebras and Classical K-Theory, Indian Statistical Institute Series. Springer, 2020, 261-279 
(with Prof. Raja Sridharan and Sunil K. Yadav).</li><br>
<li align="justify"><b>Lie commutator, solvability and nilpotency in multiplicative Lie algebras</b>, J. Algebra Appl., 20 (8), 2150138 (11 pp), 2021 (with Prof. Ramji Lal and Mani Shankar Pandey).</li><br>
<li align="justify"><b>Theory of extensions of multiplicative Lie algebras</b>, J. Lie Theory, 31 (3), 637-658, September 2021 (with Mani Shankar Pandey).</li><br>
<li align="justify"><b>Classification of multiplicative Lie algebra structures on a finite group</b>, Colloq. Math., 168, 25-34, 2022 (with Mani Shankar Pandey).</li><br>
<li align="justify"><b>Simplicity of iterated Ore extensions</b>, Asian-Eur. J. Math., 15 (3), 2250052 (11 pp), 2022 (with Mamta Balodi).</li><br>

<li align="justify"><b>The algebraic cohomotopy group and its properties</b>, Indian J. Pure Appl. Math., 54, 641-648, 2023 (with Prof. Raja Sridharan and Sunil K. Yadav).</li><br>
<li align="justify"><b>The Schur multiplier  of some finite multiplicative Lie algebras</b>, J. Lie Theory, 34 (2), 423-436, 2024 (with Amit Kumar, Deepak Pal, and Seema Kushwaha).</li><br>
<li align="justify"><b>A realistic notion of a right gyrogroup action</b>, Proc. Rom. Acad. Ser. A, 25 (1), 27-34, 2024  (with Akshay Kumar and Seema Kushwaha).</li><br>
<li align="justify"><b>Group completion of a gyrogroup</b>, Nat. Acad. Sci. Lett., in press (with Akshay Kumar, Mani Shankar Pandey, and Seema Kushwaha).</li><br>
<li align="justify"><b>Nilpotency and capability in multiplicative Lie algebras</b>, Comm. Algebra, in press  (with Amit Kumar and Mani Shankar Pandey).</li><br>
<li align="justify"><b>Some remarks on algebraic fundamental groups</b>, to appear in J. Pure Appl. Algebra (with Prof. Raja Sridharan and Sunil K. Yadav).</li><br>
<li align="justify"><b>Isoclinism in multiplicative Lie algebras</b>, to appear in Publ. Math. Debrecen  (with Mani Shankar Pandey).</li><br>

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