If no ideal in the list contains the intersection of the others, and all ideals have distinct prime radicals, the decomposition is reduced. Every primary decomposition implies a reduced primary decomposition, as follows.
Let P be the radical ideal of one of the primary ideals in the list. Let G be the intersection of all the P primary ideals in the list. As we saw in the previous section, G is primary, and rad(G) = P. Replace all the P primary ideals with G. Do this for each prime radical, and each P is represented only once.
Finally, if one of the primary ideals contains the intersection of the others, remove it, since it does not change J. Repeat this step until you have a reduced decomposition.