Home Page of Ashish K. Srivastava






Contact Information:

Assistant Professor
Department of Mathematics and Computer Science
St. Louis University, MO-63103, USA.
Office: 224 Ritter Hall
E-mail: asrivas3@slu.edu
Phone: (314)977-2848.
Fax: (314)977-1452.


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Teaching (Fall Semester, 2008)
Abstract Algebra (Math 411)


Calculus-I (Math 142)


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My Research Interests:
I am primarily interested in Noncommutative Algebra, Combinatorics and Noncommutative Geometry. My research interest includes von-Neumann regular rings, rings generated by units, group algebras of locally compact groups, Hochschild extensions of rings, and Deformation Theory of rings. I am also interested in Applied Algebra.

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Publications:

1. Essential Extensions of a Direct Sum of Simple Modules , Contemporary Mathematics, 420, Groups, Rings and Algebras, AMS (2006), 15-23, (with S. K. Jain and K. I. Beidar).

2. Right Self-Injective Rings in Which Every Element is Sum of Two Units , Journal of Algebra and Appl., Vol 6, No 2 (2007), 281-286, (with D. Khurana).

3. Unit Sum Numbers of Right Self-injective Rings , The Bulletin of Australian Mathematical Society, Vol 75, No 3 (2007), 355-360, (with D. Khurana).

4. New Characterization of $\Sigma$-injective Modules, Proc. Amer. Math. Soc., Vol 316, No 10 (2008) (with S. K. Jain and K. I. Beidar).

5. Essential Extensions of a Direct Sum of Simple Modules-II, Modules and Comodules, Trends in Mathematics, Birkhauser (2008), 243-246 (with S. K. Jain).

6. The Monochromatic Column Problem, Discrete Mathematics, Vol 308, No 17 (2008), 3906-3916 (with S. Szabo).

7. Generalized Group Algebras of Locally Compact Groups, To appear in Communications in Algebra, 2008, (with S. K. Jain and A. I. Singh).

8. On multicolour noncomplete Ramsey graphs of star graphs, To appear in Discrete Applied Mathematics, 2008, (with S. Gautam and A. Tripathi).

9. On $\Sigma$-q Rings , submitted to the Journal of Pure and Applied Algebra, (with S. K. Jain and S. Singh).

10. A Survey on Rings Generated by Units , submitted to the Annales de la Facult'e des Sciences de Toulouse Math'ematiques (volume in honor of Mel Henriksen).

11. Partial Order in a von Neumann Regular Ring , Submitted to the Communications in Algebra, (with S. K. Jain, B. Blackwood and K. M. Prasad).

12. Unit-Regular Rings Generated by Units, in progress.

13. Hochschild Extensions of Various Classes of Rings, in progress, (with Changchang Xi).

14. Rings For Which Each Simple Module is $\sigma$-injective, in progress, (with S. K. Jain and Greg Marks).

15. A Note on Unit-Central Rings, in progress, (with Greg Marks).

16. New Characterizations of $\Sigma$-injective modules-II, in progress, (with S. K. Jain).


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News

I am organizing jointly with Greg Marks a Special Session on Noncommutative Algebra in the next Joint AMS/MAA meeting at Washington DC, January 5-8, 2009. More information will be posted here later.

Conference in honor of 70th Birthday of Prof. S. K. Jain

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Talks delivered at Conferences:

1. Joint AMS-MAA Meeting, San Diego, California, USA, Jan 5-8, 2008.
2. International Conference on Rings and Things, Zanesville, June 2007.
3. Joint AMS-MAA Meeting, New Orleans, LA, USA, Jan 5-8, 2007.
4. Indo-US Ramanujan Symposium, Chennai, India, Dec 18-22, 2006.
5. Conference on Groups, Rings and Algebras at University of Wisconsin, Madison, June 10-June 12, 2005.
6. Conference on Algebra and its Applications at Ohio University, Athens, March 22-26, 2005.
7. 28 th Ohio State-Denison Mathematics Conference, 2006.

Workshops attended:

1. Special Semester on Groebner Bases and Related Methods 2006 at RICAM, Linz and RISC, Hagenberg, Austria, February 1- March 25, 2006.
2. Ph.D. School and Workshop on Noncommutative Geometry at University of Copenhagen, Denmark, Nov 2-Nov 11, 2005.


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Brief description of the work done:

1. A classical result of Zelinsky states that every linear transformation on a vector space V, except when V is one-dimensional over $Z_2$, is sum of two invertible linear transformations. In a joint work with D. Khurana, we extend this result to any right self-injective ring R by proving that every element of R is sum of two units if and only if no factor ring of R is isomorphic to $Z_2$. We have completely characterized unit sum numbers of right self-injective rings and answered a question of Henriksen. Currently, I am trying to show that every element of a unit-regular ring is a sum of two units if and only if it has no factor ring isomorphic to $Z_2$.

2. It is known that every essential extension of a direct sum of injective hulls of simple R-modules is a direct sum of injective R-modules if and only if the ring R is right noetherian. In a joint work with S. K. Jain and Kostia Beidar, we study the rings R having the property that every essential extension of a direct sum of simple R-modules is a direct sum of quasi-injective R-modules. We show that such a ring is directly finite. For a right nonsingular ring R with this property, we show that the maximal right ring of quotients is the direct product of a finite number of matrix rings over abelian regular self-injective rings. In a recent work with S. K. Jain, we show that a von-Neumann regular ring R is noetherian if and only if every essential extension of a direct sum of simple R-modules is a direct sum of quasi-injective R-modules.

3. Nakayama and Fuller showed, respectively, that over an artinian serial ring every indecomposable module is uniserial and quasi-injective and hence artinian serial rings have the property that each right ideal is a finite direct sum of quasi-injective right ideals. We call a ring with this property, a right $\Sigma$-q ring. In a joint work with S. K. Jain and Surjeet Singh, we study various classes of $\Sigma$-q rings and describe various properties of this class of rings.

4. Carl Faith defined the notion of sigma-injectivity. An injective module M is called sigma-injective if direct sum of any number of copies of M is injective. In a joint work with S. K. Jain and K. I. Beidar we give a new characterization for an injective module to be sigma-injective (Proc. Amer. Math. Soc., 2008).

5. Consider coprime positive integers $p_1, ...., p_n$ and a rectangular array of balls of m different colors with the i-th row containing $p_i$ balls of each color cyclically repeated. The problem is to find the number of columns having balls of same color. The complete solution of this question is known only for m = 2. In a joint work with Steve Szabo, we give the complete solution to the above problem for m=3.

6. Group algebras of locally compact groups have been studied by Kaplansky, Segal and many others. Alvin Hausner introduced generalized group algebras of locally compact groups. We have studied some homological properties of generalized group algebras of locally compact groups in a joint work with S. K. Jain and A. I. Singh.

7. In a joint work with S. K. Jain, B. Blackwood and K. M. Prasad, we have studied various partial orders on von Neumann regular rings.

8. With Changchang Xi, we are studying Hochschild extensions of various classes of rings.

9. In a joint work with S. K. Jain and Greg Marks, we are studying sigma-V rings.

10. In a joint work with Greg Marks, we are studying unit-central rings.


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Random Stuffs

My Erdos number is 4 along many paths.
(Erdos - Saharon Shelah - Birge Huisgen Zimmermann - S. K. Jain - Ashish K. Srivastava)
(Erdos - B. Volkmann - F. Kasch - K. I. Beidar - Ashish K. Srivastava)



Some articles about Perelman (known as the guy who doesn't care about million dollars):

The triumph of the nerd

World's top maths genius jobless .

Some Other Beautiful Articles:

The unreasonable effectiveness of mathematics - E. Wigner

There are too many B.A.D. Mathematicians - M. Henriksen