Multiplier Ring Research Articles

Similar Sublattices of Planar Lattices

AbstractThe similar sublattices of a planar lattice can be classified via its multiplier ring. The latter is the ring of rational integers in the generic case, and an order in an imaginary quadratic field otherwise. Several classes of examples are discussed, with special emphasis on concrete results. In particular, we derive Dirichlet series generating functions for the number of distinct similar sublattices of a given index, and relate them to zeta functions of orders in imaginary quadratic fields.

The exchange property for row and column-finite matrix rings

The ring B( R) of all ω× ω row-and-column-finite matrices over any (unital) von Neumann regular ring R is shown to be an exchange ring. More generally, the same is true of the multiplier ring of a non-unital regular ring with a countable unit.

Multipliers of von neumann regular rings

We analyse the structure of the multiplier ring M(R) of a(nonuni-tal)Von Neumann regular ring R. We show that M(R) is not regular in general, but every principal right ideal is generated by two idempotents. This, together with Riesz Decomposition on idempotents of M(R), furnishes a description of the monoid V(M(R)) of Murray-Von Neumann equivalence classes of idempotents which is used to effectively examine the lattice of ideals of M(R). The techniques developed here allow other applications to the category of projective modules over regular rings.

Experimental Investigation of Laser-Assisted Thermoplastic Tape Consolidation

An experimental investigation of a novel approach for manufacturing of thermoplastic matrix composites, is described. The technique is based on using laser energy as the focused heat source to melt the matrix material for subsequent consolidation, and appears to be particularly suited for thermoplastic filament winding operations. An experimental set up is defined to produce multi ply rings, and the feasibility of this technique is demon strated by discussing several samples that were produced using Ryton AC40-60 prepreg tapes. The quality of consolidation is examined through cross-sectional micrographs.

IDEALS IN COMMUTATIVE RINGS

This paper deals with one-dimensional (commutative) rings without nilpotent elements such that every ideal is generated by three elements. It is shown that in such rings the square of every ideal is invertible, i.e. divides its multiplier ring. In addition, every ideal is distinguished, in the sense that on localization at any maximal ideal it becomes either principal or dual to a principal ideal. Conversely, if a one-dimensional ring without nilpotent elements satisfies either of these conditions, and if all its residue class fields are 2-perfect and contain at least three elements, then every ideal can be generated by three elements.Bibliography: 18 titles.

Multiplier rings and primitive ideals

The multiplier ring M(A) of any ring without an identity is the biggest essential extension A--* M(A) where the image of A is an ideal (notation: A<M(A)). Recently, M(A) has received considerable attention. B. E. Johnson studied the multiplier for semigroups, rings, and topological algebras [7]. R. C. Busby used it for classifying extensions of C*-algebras in [1], and in [2] to study the spectrum of an algebra. In [6], M(A) is used from a different point of view; there the question when a C*-algebra A is an ideal in its second dual is considered. ?1 develops several useful properties of the multiplier for associative rings, including a characterization of the multiplier as the adjoint of a certain forgetful functor. Perhaps some of these considerations can be carried over to other categories, such as abelian or topological groups. In ?2 an extension A<A is considered. Not only the connection between Prim A-the primitive ideals of A-and Prim A is established, but also simultaneously and within the same framework, the correspondence between the associated regular maximal left ideals as well as the simple A and A-modules is completely described. No assumptions other than A<IA are imposed; an identity for A is not assumed. For this reason the above development might be of interest because some of the above mentioned results about A c(A had been proved previously with A=M(A) for special kinds of rings, such as C*-algebras, by using very special and frequently irrelevant properties of these rings. ?3 deals with more special extensions of the form A=S+A, with A<KA, and S a subring, where S n A ={O} is not always assumed. The first part of the paper has been written, as far as possible, so as to be selfcontained. However, in the remainder A is specialized to a C*-algebra and some familiarity with [4] (or [1] and [5]) is required. The center R of M(A) is called the centroid of A. In [4], Prim (R + A) was described. In ?4 a description of Prim M(A) is given. For M(A), just as was the case for R+A in [4], another space of ideals, obtained as the complete regularization of the primitive ones, plays an even more important role. It also is described. It is shown that the primitive ideal space of the center of A can be identified with a certain subset of ideals of the complete regularization of Prim A, thus generalizing a result of Busby [2]. The objective of the last section is to identify and characterize closed ideals A2