In mathematics, given a ring R, the K-theory spectrum of R is an Ω-spectrum K R {\displaystyle K_{R}} whose nth term is given by, writing Σ R {\displaystyle \Sigma R} for the suspension of R,

( K R ) n = K 0 ( Σ n R ) × B G L ( Σ n R ) + {\displaystyle (K_{R})_{n}=K_{0}(\Sigma ^{n}R)\times BGL(\Sigma ^{n}R)^{+}} ,

where "+" means the Quillen's + construction. By definition, K i ( R ) = π i ( K R ) {\displaystyle K_{i}(R)=\pi _{i}(K_{R})} .