<0xE2> <0x89> <0xBC> ▁a ▁ ∧ ▁a ▁ <0xE2> <0x89> <0xBC> ▁b }, ▁with ▁multiplication ▁and ▁ordering ▁based ▁on ▁K ▁and ▁ <0xE2> <0x89> <0xBC> . ▁ ▁Term in ology ▁In ▁this ▁article , ▁we ▁use ▁the ▁terms ▁defined ▁above , ▁in ▁the ▁add itive ▁notation . ▁However , ▁some ▁authors ▁use ▁alternative ▁terms : ▁▁ ▁our ▁" valu ation " ▁( s atisf ying ▁the ▁ul tr am et ric ▁inequality ) ▁is ▁called ▁an ▁" ex pon ential ▁valu ation " ▁or ▁" non - Arch im ede an ▁absolute ▁value " ▁or ▁" ult ram et ric ▁absolute ▁value "; ▁ ▁our ▁" absolute ▁value " ▁( s atisf ying ▁the ▁triangle ▁inequality ) ▁is ▁called ▁a ▁" valu ation " ▁or ▁an ▁" Arch im ede an ▁absolute ▁value ". ▁ ▁Associ ated ▁objects ▁There ▁are ▁several ▁objects ▁defined ▁from ▁a ▁given ▁valu ation ▁; ▁the ▁value ▁group ▁or ▁valu ation ▁group ▁ ▁= ▁v ( K × ), ▁a ▁subgroup ▁of ▁ ▁( though ▁v ▁is ▁usually ▁sur jective ▁so ▁that ▁ ▁= ▁ ▁); ▁the ▁valu ation ▁ring ▁R v ▁is ▁the ▁set ▁of ▁a ▁ ∈ ▁ ▁with ▁v ( a )   ≥   0 , ▁the ▁prime ▁ideal ▁mv ▁is ▁the ▁set ▁of ▁a ▁ ∈ ▁K ▁with ▁v ( a )   >   0 ▁( it ▁is ▁in ▁fact ▁a ▁maximal ▁ideal ▁of ▁R v ), ▁the ▁resid ue ▁field ▁kv ▁= ▁R v / mv , ▁the ▁place ▁of ▁ ▁associated