Ashish K. Srivastava

Cyclic modules and the structure of rings

Preface 1. Preliminaries 2. Rings characterized by their proper factor rings 3. Rings each of whose proper cyclic modules has a chain condition 4. Rings each of whose cyclic modules is injective (or CS) 5. Rings each of whose proper cyclic modules is injective 6. Rings each of whose simple modules is injective (or -injective) 7. Rings each of whose (proper) cyclic modules is quasi-injective 8. Rings each of whose (proper) cyclic modules is continuous 9. Rings each of whose (proper) cyclic modules is pi-injective 10. Rings with cyclics @0-injective, weakly injective or quasi-projective 11. Hypercyclic, q-hypercyclic and pi-hypercyclic rings 12. Cyclic modules essentially embeddable in free modules 13. Serial and distributive modules 14. Rings characterized by decompositions of their cyclic modules 15. Rings each of whose modules is a direct sum of cyclic modules 16. Rings each of whose modules is an I0-module 17. Completely integrally closed modules and rings 18. Rings each of whose cyclic modules is completely integrally closed 19. Rings characterized by their one-sided ideals References Index

RIGHT SELF-INJECTIVE RINGS IN WHICH EVERY ELEMENT IS A SUM OF TWO UNITS

A classical result of Zelinsky states that every linear transformation on a vector space V, except when V is one-dimensional over ℤ2, is a sum of two invertible linear transformations. We extend this result to any right self-injective ring R by proving that every element of R is a sum of two units if no factor ring of R is isomorphic to ℤ2.

Unit sum numbers of right self-injective rings

In a recent paper [KS] we proved that every element of a right self-injective ring R is a sum of two units if and only if R has no factor ring isomorphic to Z2 and hence the unit sum number of a nonzero right selfinjective ring is 2, ω or ∞. In this paper we characterize right self-injective rings with unit sum numbers ω and ∞. We prove that the unit sum number of a right self-injective ring R is ω if and only if R has a factor ring isomorphic to Z2 but no factor ring isomorphic to Z2 × Z2, and also in this case every element of R is a sum of either two or three units. It follows that the unit sum number of a right self-injective ring R is ∞ precisely when R has a factor ring isomorphic to Z2 ×Z2. We also answer a question of Henriksen ([H], Question E, page 192) by giving a large class of regular right self-injective rings having the unit sum number ω in which not all non-invertible elements are sum of two units. We shall consider associative rings with identity. Our modules will be unital right modules with endomorphisms acting on the left. A ring R is said to have n-sum property, for a positive integer n, if its every element can be written as a sum of exactly n units of R. It is obvious that a ring having n-sum property also has k-sum property for every positive integer k > n. The unit sum number of a ring R, denoted by usn(R), is the least integer n, if it exists, such that R has the n-sum property. If R has an element which is not a sum of units then we set usn(R) to be ∞, and if every element of R is a sum of units but R does not have n-sum property for any n, then we set usn(R) = ω. Clearly, usn(R) = 1 if and only if R has only one element. The unit sum number of a module M , denoted by usn(M), is the unit sum number of its endomorphism ring. The topic has been studied extensively (see [AV], [FS], [GMW], [GO], [GPS], [H], [Hi], [R], [U], [V], [Z]). In 1954 Zelinsky [Z] proved that usn(VD) = 2, where V is a vector space over a division ring D, unless dim(V ) = 1 and D = Z2, in which case usn(VD) = 3. As End(VD) is a (von Neumann) regular ring, in 1958 Skornyakov ([S], Problem 31, page 167) asked: Is every element of a regular ring a sum of units? This question was settled by Bergman (see [Ha]) in negative who gave an example of a regular directly-finite ring R in which 2 is invertible such that usn(R) = ∞. The result of Zelinsky has also generated a considerable interest in the unit sum numbers of right self-injective rings as End(VD) is right self-injective (see [U], [V], [R]). Recently, the authors in [KS] have characterized right self-injective rings with unit sum number 2000 Mathematics Subject Classification. 16U60, 16D50.

Shorted Operators Relative to a Partial Order in a Regular Ring

In this article, the explicit form of maximal elements, known as shorted operators, in a subring of a von Neumann regular ring, has been obtained. As an application of the main theorem, the unique shorted operator (of electrical circuits), which was introduced by Anderson–Trapp, has been derived.

On Σ-V Rings

We discuss various properties of a ring over which each simple module is Σ-injective.

On Unit-Central Rings

We establish commutativity theorems for certain classes of rings in which every invertible element is central, or, more generally, in which all invertible elements commute with one another. 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. We offer some examples of noncommutative rings in which all invertible elements commute with one another, or are central. We close with a list of problems for further research.

A review on the intensification of metal matrix composites and its nonconventional machining

Abstract Metal matrix composites (MMCs) are the new-generation advanced materials that have excellent mechanical properties, such as high specific strength, strong hardness, and strong resistance to wear and corrosion. All these qualities make MMCs suitable material in the manufacture of automobiles and aircraft. The machining of these materials is still difficult due to the abrasive nature of the reinforced particles and hardness of MMCs. The conventional machining of MMCs results in high tool wear and slow removal of materials, thereby increasing the overall machining cost. The nonconventional machining of these materials, on the contrary, ensures much better performance. This paper reviews various research works on the development of MMCs and the subsequent hybrid composites and evaluates their performances. Further, it discusses the influence of the process parameters of conventional and nonconventional machining on the performance of MMCs. At the end, it identifies the research gaps and future scopes for further investigations in this field.

DECOMPOSING ELEMENTS OF A RIGHT SELF-INJECTIVE RING

It was proved independently by both Wolfson [An ideal theoretic characterization of the ring of all linear transformations, Amer. J. Math. 75 (1953), 358-386] and Zelinsky [Every Linear Transformation is Sum of Nonsingular Ones, Proc. Amer. Math. Soc. 5 (1954), 627-630] that every linear transformation of a vector space

Influence of Abrasive Water Jet Turning Parameters on Variation of Diameter of Hybrid Metal Matrix Composite

V

New characterization of Σ-injective modules

over a division ring