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 ...

, more specifically in

algebraic K-theory Algebraic ''K''-theory is a subject area in mathematics with connections to geometry, topology, ring theory, and number theory. Geometric, algebraic, and arithmetic objects are assigned objects called ''K''-groups. These are groups in the sens ...

, the suspension

\Sigma R

of a

ring (The) Ring(s) may refer to: * Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry * To make a sound with a bell, and the sound made by a bell Arts, entertainment, and media Film and TV * ''The Ring'' (franchise), a ...

''R'' is given byWeibel, III, Ex. 1.15

\Sigma(R) = C(R)/M(R)

where

C(R)

is the ring of all infinite matrices with entries in ''R'' having only finitely many nonzero elements in each row or column and

M(R)

is its

ideal Ideal may refer to: Philosophy * Ideal (ethics), values that one actively pursues as goals * Platonic ideal, a philosophical idea of trueness of form, associated with Plato Mathematics * Ideal (ring theory), special subsets of a ring considered ...

matrices Matrix (: matrices or matrixes) or MATRIX may refer to: Science and mathematics * Matrix (mathematics), a rectangular array of numbers, symbols or expressions * Matrix (logic), part of a formula in prenex normal form * Matrix (biology), the ...

having only finitely many nonzero elements. It is an analog of

suspension Suspension or suspended may refer to: Science and engineering * Car suspension * Cell suspension or suspension culture, in biology * Guarded suspension, a software design pattern in concurrent programming suspending a method call and the calling ...

topology Topology (from the Greek language, Greek words , and ) is the branch of mathematics concerned with the properties of a Mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformat ...

. One then has:

K_i(R) \simeq K_(\Sigma R)

Notes

References

* C. Weibel
The K-book: An introduction to algebraic K-theory
Algebra Algebraic K-theory {{algebra-stub