Short definitions of the ordinals

1966 ◽  
Vol 31 (3) ◽  
pp. 409-414 ◽  
Author(s):  
Kenneth R. Brown ◽  
Hao Wang
Keyword(s):  
Von Neumann ◽  
Ordinal Numbers

In this paper, a simple inductive characterization of the ordinal numbers is stated and developed. The characterization forms the basis for a set of axioms for ordinal theory and also for several short explicit definitions of the ordinals. The axioms are shown to be sufficient for ordinal theory, and, subject to suitable existence assumptions, each of the definitions is shown to imply the axioms.The present results apply to the familiar von Neumann version of the ordinals, but the methods used are easily adapted to other versions.

scholarly journals A characterization of von Neumann rings in terms of linear systems

2016 ◽  
Vol 494 ◽  
pp. 236-244 ◽  
Author(s):  
Noemí DeCastro-García ◽  
Miguel V. Carriegos ◽  
Ángel Luis Muñoz Castañeda
Keyword(s):  
Linear Systems ◽  
Von Neumann ◽  
Von Neumann Rings

Rings of Quotients of Group Rings

1969 ◽  
Vol 21 ◽  
pp. 865-875 ◽  
Author(s):  
W. D. Burgess
Keyword(s):  
Group Ring ◽  
Group Rings ◽  
Maximum Condition ◽  
Regular Group ◽  
Von Neumann ◽  
Von Neumann Regular ◽  
Completely Reducible ◽  
Rings Of Quotients ◽  
Prime Case

The group ring AG of a group G and a ring A is the ring of all formal sums Σg∈G agg with ag ∈ A and with only finitely many non-zero ag. Elements of A are assumed to commute with the elements of G. In (2), Connell characterized or completed the characterization of Artinian, completely reducible and (von Neumann) regular group rings ((2) also contains many other basic results). In (3, Appendix 3) Connell used a theorem of Passman (6) to characterize semi-prime group rings. Following in the spirit of these investigations, this paper deals with the complete ring of (right) quotients Q(AG) of the group ring AG. It is hoped that the methods used and the results given may be useful in characterizing group rings with maximum condition on right annihilators and complements, at least in the semi-prime case.

Monotone convergence theorems for semi-bounded operators and forms with applications

2010 ◽  
Vol 140 (5) ◽  
pp. 927-951 ◽  
Author(s):  
Jussi Behrndt ◽  
Seppo Hassi ◽  
Henk de Snoo ◽  
Rudi Wietsma
Keyword(s):  
Differential Operators ◽  
Multiplication Operators ◽  
Monotone Convergence ◽  
Monotone Sequence ◽  
Bounded Operators ◽  
Von Neumann ◽  
Sturm Liouville ◽  
Adjoint Operators ◽  
Liouville Operators

AbstractLet Hn be a monotone sequence of non-negative self-adjoint operators or relations in a Hilbert space. Then there exists a self-adjoint relation H∞ such that Hn converges to H∞ in the strong resolvent sense. This result and related limit results are explored in detail and new simple proofs are presented. The corresponding statements for monotone sequences of semi-bounded closed forms are established as immediate consequences. Applications and examples, illustrating the general results, include sequences of multiplication operators, Sturm–Liouville operators with increasing potentials, forms associated with Kreĭn–Feller differential operators, singular perturbations of non-negative self-adjoint operators and the characterization of the Friedrichs and Kreĭn–von Neumann extensions of a non-negative operator or relation.

Extending the Property of a Maximal Right Ideal

2006 ◽  
Vol 13 (01) ◽  
pp. 163-172 ◽  
Author(s):  
Gary F. Birkenmeier ◽  
Dinh Van Huynh ◽  
Jin Yong Kim ◽  
Jae Keol Park
Keyword(s):  
Direct Summand ◽  
The Self ◽  
Von Neumann ◽  
Von Neumann Regular Rings ◽  
Von Neumann Regular ◽  
Regular Rings

We extend various properties from a direct summand X of a module M, whose complement is semisimple, to its trace in M or to M itself. The case when MR = RR and the properties are injectivity or P-injectivity is fully described. As applications, we extend some known results for right HI-rings and give a new characterization of semisimple rings. We conclude this paper by giving some conditions that yield the self-injectivity of von Neumann regular rings.

scholarly journals Characterization of Lie derivations on von Neumann algebras

2013 ◽  
Vol 438 (1) ◽  
pp. 533-548 ◽  
Author(s):  
Xiaofei Qi ◽  
Jinchuan Hou
Keyword(s):  
Von Neumann Algebras ◽  
Von Neumann ◽  
Lie Derivations
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