In mathematics, given a

ring Ring 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 :(hence) to initiate a telephone connection Arts, entertainment and media Film and ...

''R'', the ''K''-theory spectrum of ''R'' is an

Ω-spectrum In algebraic topology, a branch of mathematics, a spectrum is an object representing a generalized cohomology theory. Every such cohomology theory is representable, as follows from Brown's representability theorem. This means that, given a cohomolo ...

K_R

whose ''n''th term is given by, writing

\Sigma R

for the

suspension Suspension or suspended may refer to: Science and engineering * Suspension (topology), in mathematics * Suspension (dynamical systems), in mathematics * Suspension of a ring, in mathematics * Suspension (chemistry), small solid particles suspend ...

of ''R'', :

(K_R)_n = K_0(\Sigma^n R) \times BGL(\Sigma^n R)^+

, where "+" means the Quillen's + construction.Dominique Arlettaz, Algebraic ''K''-theory of rings from a topological view poin

/ref> By definition,

K_i(R) = \pi_i(K_R)

References

Algebraic K-theory {{topology-stub