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

${\displaystyle (K_{R})_{n}=K_{0}(\Sigma ^{n}R)\times BGL(\Sigma ^{n}R)^{+}}$,

where "+" means the Quillen's + construction. ^[1] By definition, ${\displaystyle K_{i}(R)=\pi _{i}(K_{R})}$.

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