Let $R$ be a unital associative ring, let $I \trianglelefteq R$ be a two-sided ideal, and let $\pi:R\to R/I$ be the quotient homomorphism. Define the relative K-theory object $K(R,I)$ to be the homotopy fibre of the induced map of algebraic K-theory spectra $K(\pi):K(R)\to K(R/I)$, and write $K_i(R,I)=\pi_i K(R,I)$ for $i=0,1$. Then there is an exact sequence

\begin{align*} K_1(R,I) \longrightarrow K_1(R) \longrightarrow K_1(R/I) \xrightarrow{\partial} K_0(R,I) \longrightarrow K_0(R) \longrightarrow K_0(R/I). \end{align*}

Here $\partial$ is the connecting homomorphism associated to the homotopy fibre sequence $K(R,I)\to K(R)\to K(R/I)$.