SBI ring

In algebra, an SBI ring is a ring R (with identity) such that every idempotent of R modulo the Jacobson radical can be lifted to R. The abbreviation SBI was introduced by Irving Kaplansky and stands for "suitable for building idempotent elements" (Jacobson 1956, p.53).

Examples

Any ring with nil radical is SBI.
Any Banach algebra is SBI: more generally, so is any compact topological ring.
The ring of rational numbers with odd denominator, and more generally, any local ring, is SBI.

gollark: Are you suggesting Assembly is fine for webapps too?

gollark: I don't really believe that.]

gollark: The "wrong"ness of opinions, I guess, depends if your disagreement is based on aesthetic preference differences, or wrong facts/logic.

gollark: Hey, if you think the argument of popularity is fine applied to PHP, I can apply it to opinions.

gollark: Like I said, if I say "assembly is worse than PHP for making web applications", most people will say "yes, that is a fact".

References

Jacobson, Nathan (1956), Structure of rings, American Mathematical Society, Colloquium Publications, 37, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-1037-8, MR 0081264, Zbl 0073.02002
Kaplansky, Irving (1972), Fields and Rings, Chicago Lectures in Mathematics (2nd ed.), University Of Chicago Press, pp. 124–125, ISBN 0-226-42451-0, Zbl 1001.16500

This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.