In a 1957 paper (http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.ijm/1255378502), Tate shows that if $I \subset R$ is an ideal of the noetherian ring R then there is a graded commutative DGA $X$ over $R$ with $H_i X=0$ except $H_0 X= R/I$ (I guess R should be noetherian). Further, $X$ is a free $R$ module in each degree. Is it known if there is a similar result for any other classes of commutative $R$ algebra?
If such an extension of the result is false, I would very much appreciate a counterexample.
Thanks for your time, Sean
EDIT: (by other classes I mean commutative R algebras (Not DGAs) that are not of the form $R/I$). Also, I am happy with counter examples where the DGA is not level-wise free but instead projective or flat (although it shouldn't end up mattering for my purposes).
I am not concerned with relaxing the noetherian hypothesis but in resolving a different commutative $R$-algebra (not in the sense of Tate).
An extension of the above result would be equivalent to saying that there are no level wise free/projective/flat $E_\infty$ algebras with homology a given commutative $R$-algebra $A$ that are not strictly commutative (in the graded sense).
I worry that I am making things more confusing, sorry if that is the case, and thank you for the answers so far.