We define the Artinian and Noetherian algebra which consist of formal series involving exponents which are not necessarily integers. All of the usual operations are defined here and characterized. As an application, we compute the algebra of symmetric functions with nonnegative real exponents. The applications to logarithmic series and the Umbral calculus are deferred to another paper. On d��finit ici les alg��bres Artinienne et Noetherienne comme ��tant des alg��bres constitu��es des s��ries formelles �� exposants pas n��cessairement entiers. On definit sur ces alg��bres toutes les op��rations classiques et on les caracterise. Comme exemple d'exploitation de cette th��orie, on s'interesse �� alg��bre de fonctions sym��triques &agrave exponsants r��els en nonn��gatifs. Une autre publication est consacr��e aux applications aux series logarithmiques et au calcul ombral.