LetHbe a Hopf algebra over a fieldk:, andAanH-module algebra, with subalgebra ofH-invariants denoted byAH. When (H, R) is quasitriangular andAis quantum commutative with respect to (H,R), (e.g. quantum planes, graded commutative superalgebras), thenAH⊂ center ofA=Z(A). In this paper we are mainly concerned with actions ofHfor whichAH⊂Z(A). We show that under this hypothesis there exists strong relations between the ideal structures ofAHAandA#H.