The Grothendieck group K₀ of an arbitrary csp-ring

Fix an infinite set L of primes. For every p e L, let R_p be either the ring of p-adic integers or the residue class ring Z /p^kZ (the number k > 0 may depend on p). Define K = ПRp and T =0Rpс K ; p^eL peL it is clear that T is an ideal of the ring K. By a csp-ring we mean any subring R of the ring K such that T с R and the quotient ring R /T is a field. The symbol K₀(R) denotes the Grothendieck group of the monoid of isomorphism classes of finitely generated projective modules over R (with direct sum as the operation). We find necessary and sufficient conditions for a module over R to be a finitely generated projective module. These conditions enable us to prove the following theorem. Theorem 7. For every csp-ring R, the Grothendieck group K₀(R) is a free group of countable rank. If we have two csp-rings R and S, then every ring homomorphism R ^ S induces a group homomorphism K₀(R) ^ K₀(S ). We describe this group homomorphism for arbitrary csp-rings R and S.

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