1. A module is called automorphism-invariant if it is invariant under any automorphism of its injective hull. In a joint work with Pedro Guil Asensio we show that automorphism-invariant modules satisfy the exchange property. We further show that these modules provide a new class of clean modules. We extend further the result of Dickson and Fuller and show that if R is any algebra over a field with more than two elements, then a module over R is automorphism-invariant if and only if it is quasi-injective. In a joint work with Noyan Er and Surjeet Singh, we show that a simple right automorphism-invariant ring is right self-injective, thus answering in the affirmative an open question due to John Clark and Dinh van Huynh. We also show that automorphism-invariant modules coincide with pseudo-injective modules.
2. One of the most interesting results in ring theory and homological algebra is a result due to Bass, Cartan-Eilenberg,Paap, Matlis which states that a ring R is right noetherian if and only if every direct sum of injective right R-modules is injective. A module M is called $\Sigma$-injective if direct sum of any number of copies of M is injective. So it follows easily that a ring R is right noetherian if and only if every injective right R-module is $\Sigma$-injective. In a joint work with Jain and Beidar we extend this result and show that a ring R is right noetherian if and only if for each injective right R-module M, every essential extension of a countable direct sum of copies of M is a direct sum of injective modules. In a joint work with Guil-Asensio and Jain, we extend it further and show that a ring R is right noetherian if and only if for each injective module M, every essential extension of a countable direct sum of copies of M is a direct sum of modules that are either injective or projective.
3. A classical result of Zelinsky and Wolfson states that every linear transformation on a vector space V, except when V is one-dimensional over field of two elements, is a sum of two invertible linear transformations. In a joint work with Khurana, we extend this result to any right self-injective ring R by proving that every element of R is the sum of two units if and only if no factor ring of R is isomorphic to field of two elements. In a survey paper on rings generated by units, I have proposed several interesting conjectures. Recently, in a joint work with Feroz Siddique, we prove that if R is a right self-injective ring, then for every element x in R there exists a unit u in R such that both x+u and x-u are units in R if and only R has no factor isomorphic to Z/2Z or Z/3Z.
4. In a joint work with S. Singh, we propose the dual notion of automorphism-invariant modules and call them dual automorphism-invariant modules. We give various examples of dual automorphism-invariant module and study its properties. In particular, we study abelian groups and prove that dual automorphism-invariant abelian groups must be reduced. We show that over a right perfect ring R, a lifting right R-module M is dual automorphism-invariant if and only
if M is quasi-projective.
5. Right $\Sigma$-V rings were introduced by Goursaud and Velette in 1975 but it seems that these rings have been forgotten by the community of ring theorists and module theorists. We have proved that right $Sigma$-V rings are always directly finite and if, in addition, they are right non-singular then they have a bounded index of nilpotence. My paper on right $\Sigma$-V rings proposes many interesting open problems.
6. In a joint work with S. K. Jain and S. Singh, we have introduced the notion of $\Sigma$-q rings. We call a ring whose each right ideal is a finite direct sum of quasi-injective right ideals, 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.
7. 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 joint works with Jain and 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 a semi-regular ring R with this property is noetherian.
8. 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.
9. 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 with applications to electrical circuits.
10. In a joint work with Greg Marks and D. Khurana, we have studied rings whose each unit element is central. We have called such rings as unit-central rings. We prove that if R is a semiexchange
ring (i.e. its factor ring modulo its Jacobson radical is an exchange
ring) with all invertible elements central, then R is commutative. We also
prove that if R is a semiexchange ring in which all invertible elements commute with one another, and R has no factor ring with two elements, then R
is commutative.
11. 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 monochromatic column problem 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.
12. In a joint work with Gautam and Tripathi, we study multicolor noncomplete Ramsey graphs of star graphs. We compute the value of the graph Ramsey number R(K(1,n1),K(1,n2),..., K(1,nk)).
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