Let A: be a p-adic field and let G be a reductive group defined over k.Let G be a semigroup in <#, i.e. a multiplicative subset with the same unity as G We shall assume that there exists an open compact subgroup A of à which is contained in G. Let âë(G 9 A) be the free Z-module generated by the double cosets of G modulo A, with a product defined as in [3, Lemma 6].We have an associative ring with unity which we shall call the Hecke Ring of G with respect to A. Let A 0 be a normal subgroup of A satisfying our conditions H-l and H-2 of §1.Our purpose is to find generators and relations for 0t(G 9 A 0 ) = 0t.There exists a finitely generated polynomial ring Z\G~\ which together with the group ring Z[A/A 0 ] generates 0t\ moreover 0t is a Z[A/A 0 ]-bimodule having Z\p\ as basis.Our hypothesis H-l and H-2 are verified for the principal congruence subgroups of most of the classical groups considered in [2].We thank Mr. J. Shalika for the helpful discussions during the preparation of this work and also for pointing out some hopes that this might bring in solving Harish-Chandra conjecture on the finite dimensionality of the irreducible continuous representations of these rings.1. General results.Let J be a connected A:-closed subgroup of G consisting only of semisimple elements, and N + and N~ be maximal kclosed unipotent subgroups normalized by T. We set N + = N + n A, and U~ = TV" n A. We shall now state our first condition:Condition H-l.There exists a finitely generated semigroup D in T such that G = ADA (disjoint union), and for all del) we have dU + d~x c fZ+andrf-^-rfc: £/".We turn now to our second condition.We let A 0 be a normal subgroup of A and we set U£ = U + n A 0 and Uö = U~ n A 0 .We shall assume that T N + n A 0 = (TnA 0 ) • U%.Condition H-2.There exists a semigroup D in T such that A 0 = UQ VUÖ for a certain subgroup V of A 0 normalized by Z), and for all d in D we havedl/Jd" 1 c l/Jandd^É/ödcz U^.Let us denote by 1 the unity of 0t and by g the double coset A 0 gA 0 .We shall denote the product in 01 by *.THEOREM 1. Condition H-2 implies that D = A 0 DA 0 is a semigroup in G and 010, A 0 ) ~ Z[D~].