1. The problem statement, all variables and given/known data My textbook says that A[X, Y]/(XY) is a subring of A[X]⊕A[Y], but aren't they isomorphic? (A is any commutative ring) 2. Relevant equations 1st Ismorphism Theorem 3. The attempt at a solution I can construct the map φ: A[X, Y] → A[X]⊕A[Y] f(X)+g(Y)+h(X, Y)*X*Y → f(X)+g(Y), this map is surjective (since the inverse of any f(X)+g(Y) is f(X)+g(Y)+(XY)) Ker(φ)=(XY), then by the 1st isomorphism theorem: A[X, Y]/(XY)≅A[X]⊕A[Y], am I stupid?