Suspension of a ring

In algebra, more specifically in algebraic K-theory, the suspension ${\displaystyle \Sigma R}$ of a ring R is given by^[1] ${\displaystyle \Sigma (R)=C(R)/M(R)}$ where ${\displaystyle C(R)}$ is the ring of all infinite matrices with entries in R having only finitely many nonzero elements in each row or column and ${\displaystyle M(R)}$ is its ideal of matrices having only finitely many nonzero elements. It is an analog of suspension in topology.

One then has: ${\displaystyle K_{i}(R)\simeq K_{i+1}(\Sigma R)}$.

References

1. Weibel, III, Ex. 1.15

• C. Weibel "The K-book: An introduction to algebraic K-theory"