Let R the polynomial ring in n variables with complex coefficients and I an ideal of R. Is it true that if R/I is CM also R/J is CM (where J is the radical of I)? Is there a relations between a resolution of R/J and one of R/I? What if I suppose that proj.dim(R/I)=2?
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It is not true, but the example is not easy to find $I = (x_2^2x_4x_5,x_1x_3x_3x_4, x_3x_4x_1x_5)$! 


Yes. From Eisenbud's Commutative Algebra: a ring $S$ is CohenMacaulay iff all the maximal ideal $m$ of $S$ satisfies codim($m$) = depth($m$). Now, the maximal ideals of $R/J$ are the same as $R/I$ and their depths and codimensions are the same as well. 

