Wolfram Demonstrations Project

13,000+ Open Interactive Demonstrations
Powered by Notebook Technology »

Equivalence of Projections in Involutive Rings

Initializing live version
Download to Desktop

Requires a Wolfram Notebook System

Interact on desktop, mobile and cloud with the free Wolfram Player or other Wolfram Language products.

Let be a ring with involution , that is, . An element is called a projection if it is self-adjoint () and idempotent (). The projections and are said to be equivalent, written , when exists such that and . Projections are algebraically equivalent if there exist and such that and .

[more]

Let be the involutive ring of matrices over , the field of three elements, with the matrix transpose as involution. The set of all projections in is where

, .

This Demonstration shows that and are algebraically equivalent, but not equivalent. In this case, the system of equations and has 10 solutions, but not one of them is of the form .

[less]

Contributed by: Izidor Hafner (March 2018)
Open content licensed under CC BY-NC-SA

Snapshots

Details

This example and definitions were taken from [1, pp. 3–10].

References

[1] S. K. Berberian, Baer *-Rings, New York: Springer-Verlag 1972.

[2] I. Kaplansky, Rings of Operators, New York: W. A. Benjamin, 1968.

Embed Code

Take advantage of the Wolfram Notebook Emebedder for the recommended user experience.

Related Demonstrations More by Author
Browse all topics
Feedback (field required)
Email (field required) Name
Occupation Organization
Note: Your message & contact information may be shared with the author of any specific Demonstration for which you give feedback.
Send