General Topology-I

This text, drawn from the author's lectures at the University of California at Berkeley, is intended as a textbook for a one-term course in basic ring theory. The material covered includes the Wedderburn-Artin theory of semi-simple rings, Jacobson's theory of the radical representation theory of groups and algebras, prime and semi-prime rings, primitive and semi-primitive rings, division rings, ordered rings, local and semi-local rings, and perfect and semi-perfect rings. By aiming the level of writing at the novice rather than at the expert, and by stressing the role of examples and motivation, the author has produced a text which is suitable not only for use in a graduate course, but also for self-study by other interested graduate students. Numerous exercises are also included. This graduate textbook on rings, fields and algebras is intended for graduate students in mathematics.

A first course in noncommutative rings, by T. Y. Lam. Pp. 385. £37 (pb), £62.50 (hb). 2001. ISBN 0 387 95325 6 (pb), 0 387 95183 0 (hb) (Springer-Verlag).

This paper introduces five notions, including π-algebras, π-ideals, Hopf π-algebras, π-modules and Hopf π-modules, verifies the fundamental isomorphism theorem of π-algebras and studies some algebraic properties of Hopf π-algebras as well.

Hopf π- Algebras

Uncertainty relations play a central role in quantum mechanics. Entropic uncertainty relations in particular have gained significant importance within quantum information, providing the foundation for the security of many quantum cryptographic protocols. Yet, little is known about entropic uncertainty relations with more than two measurement settings. In the present survey, we review known results and open questions.

https://iopscience.iop.org/article/10.1088/1367-2630/12/2/025009/pdf

Entropic uncertainty relations—a survey

Commutative rings I

Lifting Modules: Supplements and Projectivity in Module Theory

It is well-known that a ring Ris semiperfect if and only if RR (orRR ) is a supplemented module. Considering weak supplementsinstead of supplements we show that weakly supplemented modules Mare semilocal (i.e.M/Rad(M) is semisimple) and that R is a semilocal ring if and only if RR (orRR ) is weakly supplemented. In this context the notion of finite hollow dimension (or finite dual Goldie dimension) of modules is of interest and yields a natural interpretation of the Camps-Dicks characterization of semilocal rings. Finitely generated modules are weakly supplemented if and only if they have finite hollow dimension (or are semilocal).

On semilocal modules and rings

This dissertation reviews attempts of dualizing the Goldie dimension. Moreover we choose one of these attempts as the dualization of Goldie dimension and study modules with this finiteness condition under various aspects. Chapter 1 defines basic ideas as dualizations of well-known notions. Small sub-modules, hollow modules, small covers, supplements, coclosed submodules and coin-dependent families of submodules are introduced as dual concepts of essential sub-modules, uniform modules, essential extensions, complements, closed submodules and independent families of submodules. In Chapter 2 existing attempts of dualizing the Goldie dimension are reviewed and compared. Section 2.1 is devoted to the earliest approach while in section 2.2 three equivalent approaches are considered. Section 2.3 states a general lattice theoretical approach equivalent to the approaches in 2.2. The core of this dissertation is formed by Chapter 3. In Section 3.1 we choose one of the approaches as dualization of Goldie dimension and call it hollow dimension. The main characterizations and properties are stated. Dimension formulas as for vector spaces are considered in Section 3.2. We show in Section 3.3 that rings with finite hollow dimension are exactly the semilocal rings. The situation when the hollow dimension of a module coincides with the hollow dimension of the endo-morphism ring is studied in Section 3.4. Here we study properties of modules with semilocal endomorphism rings as well. Relationships of certain chain conditions and hollow dimension are stated in Section 3.5. In Section 3.6 modules with property AB5* whose submodules have finite hollow dimension are considered. The dual concept of extending (or CS) modules namely lifting modules is introduced in Chapter 4 and their relation to hollow modules is studied. Basic definitions of lifting modules and a decomposition of lifting modules with finite hollow dimension are given in Section 4.1. The structure of lifting modules with certain chain conditions on the radical is given in Section 4.2. In the last chapter of this thesis, Chapter 5, we study dualizations of singular and polyform modules in connection with Goldie's Theorem. The notion of M-small and non-M-small modules are introduced in Section 5.1 as dual concepts of M-singular and non-M-singular modules. Eventually co-rational submodules and copolyform modules are defined in Section 5.2 as dual notions of rational submodules and polyform modules.

On Dual Goldie Dimension

It follows from a recent paper by Ding and Wang that any ring which is generalized supplemented as left module over itself is semiperfect. The purpose of this note is to show that Ding and Wang's claim is not true and that the class of generalized supplemented rings lies properly between the class of semilocal and semiperfect rings. Moreover we rectify their claim by introducing a wider notion of local submodules.

On a recent generalization of semiperfect rings

Primeness on modules can be defined by prime elements in a suitable partially ordered groupoid. Using a product on the lattice of submodules ℒ(M) of a module M defined in [3] we revise the concept of prime modules in this sense. Those modules M for which ℒ(M) has no nilpotent elements have been studied by Jirasko and they coincide with Zelmanowitz' "weakly compressible" modules. In particular we are interested in representing weakly compressible modules as a subdirect product of "prime" modules in a suitable sense. It turns out that any weakly compressible module is a subdirect product of prime modules (in the sense of Kaplansky). Moreover if M is a self-projective module, then M is weakly compressible if and only if it is a subdirect product of prime modules (in the sense of Bican et al.). An application to Hopf actions is given.

/pdf/prime-elements-in-partially-ordered-groupoids-applied-to-5dvz5765dd.pdf

Christian Lomp

Papers

Lifting Modules: Supplements and Projectivity in Module Theory

On semilocal modules and rings

On Dual Goldie Dimension

On a recent generalization of semiperfect rings

Prime elements in partially ordered groupoids applied to modules and hopf algebra actions