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.

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