▁fields . ▁ ▁Over ▁a ▁discrete ▁topological ▁ring , ▁the ▁ring ▁of ▁restricted ▁power ▁series ▁coinc ides ▁with ▁a ▁polynomial ▁ring ; ▁thus , ▁in ▁this ▁sense , ▁the ▁notion ▁of ▁" rest rict ed ▁power ▁series " ▁is ▁a ▁general ization ▁of ▁a ▁polynomial . ▁ ▁Definition ▁ ▁Let ▁A ▁be ▁a ▁linear ly ▁top olog ized ▁ring , ▁separated ▁and ▁complete ▁and ▁ ▁the ▁fundamental ▁system ▁of ▁open ▁ide als . ▁Then ▁the ▁ring ▁of ▁restricted ▁power ▁series ▁is ▁defined ▁as ▁the ▁project ive ▁limit ▁of ▁the ▁polynomial ▁rings ▁over ▁: ▁. ▁In ▁other ▁words , ▁it ▁is ▁the ▁completion ▁of ▁the ▁polynomial ▁ring ▁ ▁with ▁respect ▁to ▁the ▁fil tr ation ▁. ▁Sometimes ▁this ▁ring ▁of ▁restricted ▁power ▁series ▁is ▁also ▁denoted ▁by ▁. ▁ ▁Clear ly , ▁the ▁ring ▁ ▁can ▁be ▁identified ▁with ▁the ▁sub ring ▁of ▁the ▁formal ▁power ▁series ▁ring ▁ ▁that ▁consists ▁of ▁series ▁ ▁with ▁coefficients ▁; ▁i . e ., ▁each ▁ ▁contains ▁all ▁but ▁finit ely ▁many ▁coefficients ▁. ▁Also , ▁the ▁ring ▁satisfies ▁( and ▁in ▁fact ▁is ▁character ized ▁by ) ▁the ▁universal ▁property : ▁for ▁( 1 ) ▁each ▁continuous ▁ring ▁hom omorphism ▁ ▁to ▁a ▁linear ly ▁top olog ized ▁ring ▁, ▁separated ▁and ▁complete ▁and ▁( 2 ) ▁each ▁elements ▁ ▁in ▁, ▁there ▁exists ▁a ▁unique ▁continuous ▁ring ▁hom omorphism ▁ ▁extending ▁. ▁ ▁T ate ▁algebra ▁ ▁In ▁rig id ▁analysis , ▁when ▁the ▁base ▁ring ▁A ▁is ▁the ▁valu ation ▁ring ▁of ▁a ▁complete