  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\'efinit ici les alg\`ebres Artinienne et Noetherienne comme \'etant des
alg\`ebres constitu\'ees des s\'eries formelles \`a exposants pas
n\'ecessairement entiers. On definit sur ces alg\`ebres toutes les op\'erations
classiques et on les caracterise. Comme exemple d'exploitation de cette
th\'eorie, on s'interesse \`a alg\`ebre de fonctions sym\'etriques &agrave
exponsants r\`eels en nonn\'egatifs. Une autre publication est consacr\'ee aux
applications aux series logarithmiques et au calcul ombral.
