algebra Algebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic ope ...

, an SBI ring is a ring ''R'' (with identity) such that every idempotent of ''R''

modulo In computing and mathematics, the modulo operation returns the remainder or signed remainder of a division, after one number is divided by another, the latter being called the '' modulus'' of the operation. Given two positive numbers and , mo ...

the Jacobson radical can be lifted to ''R''. The abbreviation SBI was introduced by Irving Kaplansky and stands for "suitable for building idempotent elements".

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 number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (for example, The set of all ...

s with odd denominator, and more generally, any

local ring In mathematics, more specifically in ring theory, local rings are certain rings that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on algebraic varieties or manifolds, or of ...

, is SBI.

Citations

References

* * Ring theory {{abstract-algebra-stub