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When I use several ideals, I don't have commutativity. Is this normal?

R.<x,a,b>=QQ[]

P1=(X-a)*(X-b)

I=R.ideal(a+b)

J=R.ideal(a*b)

A=P.mod(I).mod(J) => X^2 - b^2

B=P.mod(J).mod(I) => X^2

A==B; => False

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