SCHUR RING AND QUASI — SIMPLE MODULES PEDRO

Let R be a ring of algebraic integers of an algebraic number field F and let G ≤ GLn(R) be a finite group. In this paper we show that the R-span of G is just the matrix ring Mn(R) of the n× n-matrices over R if and only if G/Opi(G) is absolutely simple for all pi ∈ π, where π is the set of the positive prime divisors of |G| and Opi(G) is the largest normal pi-subgroup. Subjclass : Primary 20C20 ; Secondary 19A22.


Introduction.
Let F be an algebraic number field with ring of algebraic integers R and let π = {p 1 , ..., p t } be a set of positive prime numbers.Assume that the I i are maximal ideals of R such that p i ∈ I i (i = 1, . . ., t).
Then R π denotes a localization of R at L π .Thus R π is a principal ideal having quotient field of characteristic zero and containing a unique prime ideal I i such that p i ∈ I i , i = 1, . . ., n.We denote the Jacobson radical of R π be J(R π ).Therefore the residue ring K m = R π /J(R π ) is a semi-simple ring of characteristic m.We may write where the k i are fields of characteristic p i (i = 1, . . ., t).
If G is a finite group then we obtain by (1.1).
From (1.2) it follows that where the f i are orthogonal central idempotents in Observe that R i is isomorphic to Rv p i , where Rv p i is a complete valuation ring corresponding to the discrete valuation v p i associated to the maximal ideal I i of R. Thus, we may write where the fi are orthogonal central idempotents in Rπ G.
Let π l be any set of positive prime numbers.Assume that R π l is the localization of R at L π l .Then we have In the study of the Schur ring M n (R) of a commutative ring R the main problem is to find a finite group G ≤ GL n (R) such that the R-span of G coincides with the matrix ring M n (R) of the n × n-matrices on K.The more precise question, in general sense, which Azumaya algebras over R are obtainable as an epimorphic image of the group-ring RG for some finite group G.

Notations and Definitions.
Throughout the paper F denote an algebraic number field and R denote the ring of algebraic integers of F .Moreover, K m is semi-simple ring of characteristic m with maximal ideals K i and residue fields k i = K m /K i of characteristic p i .For an maximal ideal I of R we denote by φ the I-adic valuation on F .Here R vp denote the valuation ring of v p and Rvp denote the complete valuation ring corresponding to the discrete valuation v p .Let M n (R) stand for the ring of (n × n)-matrices over R. We write GL n (R) for the multiplicative group of the invertible elements of M n (R).For a finite subgroup G of GL n (R) we let hGi R be the R-span of G in M n (R).Let π be a set of natural primes.We denote the fields of rational and complex numbers by Q and C, respectively.

Preliminary Results.
Let R be a commutative ring and let G ≤ GL n (R) be a finite group.Then the matrix ring M n (R) is called Schur ring if hGi R = M n (R).
Lemma 2.0.1.Let k be a field of characteristic p and let G ≤ GL n (k) be a finite group.Assume that V is the kG-module corresponding to G. Then G is absolutely simple if and only if hGi k = M n (k).

Proof.
If G is absolutely simple then by Burnside's theorem the assertion follows.Conversely, let Ġ be a finite group with a representation ϕ : Proof.We have Therefore hGi k i = M n (k i ).Thus applying the last lemma the result follows.Conversely, applying again the lemma (2.0.1) we deduce that for all i.Thus we may write