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In the first part of this paper we present explicit formulas for primitiveidempotents in arbitrary Frobenius algebras using the entries of representingmatrices coming from projective indecomposable modules with respect to acertain choice of basis. The proofs use a generalisation of the well knownFrobenius-Schur relations for semisimple algebras.  The second part of this paper considers $\Oh$-free $\Oh$-algebras of finite$\Oh$-rank over a discrete valuation ring $\Oh$ and their decomposition mapsunder modular reduction modulo the maximal ideal of $\Oh$, thereby studying themodular representation theory of such algebras.  Using the formulas from the first part we derive general criteria for such adecomposition map to be an isomorphism that preserves the classes of simplemodules involving explicitly known matrix representations on projectiveindecomposable modules.  Finally we show how this approach could eventually be used to attack aconjecture by Gordon James in the formulation of Meinolf Geck forIwahori-Hecke-Algebras, provided the necessary matrix representations onprojective indecomposable modules could be constructed explicitly.

Formulas for primitive idempotents in Frobenius algebras and an application to decomposition maps