Finite Complete Intersection Algebras and the Completeness Radical, Original Research Article

LetAbe a ring, and letBbe a finiteA-algebra. IfBis of the formA[X1,…,Xn]/(f1,…,fn) then we say thatBis a complete intersection overA. We show that such an algebra is projective as anA-module and Gorenstein as anA-algebra. Under the condition thatAis noetherian we show that the finiteA-algebras of the formA[[X1,…,Xm]]/(g1,…,gm) are exactly those complete intersectionsBoverAfor which one hasimage. Here cr(A) denotes the largest ideal ofAwith respect to whichAis complete. This ideal, which we call the completeness radical ofA, satisfies the usual radical axioms.