K0 of a ring ($K_0(A)$)

The Grothendieck group of finitely generated projective right $A$-modules under direct sum. An element is a formal difference $[P]-[Q]$. Equality of two positive classes means $P\oplus T\cong Q\oplus T$ for some projective $T$; adding a complement of $T$ converts this into an isomorphism after a common finite free summand. This stabilization criterion is essential in patching arguments.