K_0
Over a field, every finitely generated module is a vector space, completely described by one number — its dimension. Over a general ring, the well-behaved modules to count are the projective ones (the direct summands of free modules), and a single dimension no longer captures them. K_0 of a ring is the universal accounting device for these modules: it records how projective modules add up under direct sum, with formal subtraction allowed. It is the first and most computable rung of algebraic K-theory.
For a ring R, let P(R) be the commutative monoid of isomorphism classes of finitely generated projective right R-modules, with addition given by direct sum. Then K_0(R) is the Grothendieck group of this monoid. Every element of K_0(R) is a formal difference [P] - [Q] of classes of projectives, and [P] = [Q] in K_0(R) precisely when P and Q are stably isomorphic, meaning P ⊕ R^n is isomorphic to Q ⊕ R^n for some n. A ring homomorphism R -> S induces a homomorphism K_0(R) -> K_0(S) by extension of scalars, making K_0 a functor.
When R is a field or a principal ideal domain, every finitely generated projective is free, so K_0(R) is Z, generated by the class of R itself. For a Dedekind domain such as a ring of integers, K_0(R) is Z ⊕ Cl(R), where Cl(R) is the ideal class group: the rank lands in Z and the class group records how projective rank-one modules (fractional ideals) fail to be free. Thus K_0 already encodes a classical arithmetic invariant.
For R = Z[sqrt(-5)], the ring of integers of Q(sqrt(-5)), the class group has order 2, so K_0(R) = Z ⊕ Z/2Z. The nonprincipal ideal (2, 1 + sqrt(-5)) is a finitely generated projective module that is not free, and its class is the nonzero element of the Z/2Z summand.
K_0 detects the class number 2 of Z[sqrt(-5)].
The reduced group, often written K_0-tilde, is the cokernel of Z -> K_0(R) sending 1 to [R]; it measures projectives that are not stably free. For commutative R, the rank map K_0(R) -> H^0(Spec R, Z) splits off a copy of the continuous integer-valued functions on the spectrum, and the reduced part is the Picard-type information.