Paul Melvin proved that 0-concordant 2-knots have diffeomorphic Gluck twists, but until recently there were no known proofs that there is more than one 0-concordance class. Now Sunukjian and Dai-Miller have found many examples using Heegaard Floer technology applied to the Seifert 3-manifolds which the 2-knots bound. In this talk we give another proof using Alexander ideals. The main theorem is that the Alexander ideal induces a homomorphism from the 0-concordance monoid of 2-knots to the ideal class monoid of Z[t,t−1]. A corollary is that any 2-knot with nonprincipal Alexander ideal cannot be 0-slice, and moreover has no inverse in the 0-concordance monoid. This is the first proof that the monoid is not a group, and gives another proof of the existence of infinitely many linearly independent 0-concordance classes. These techniques also apply to higher genus surfaces, where we give the first results on 0-concordance. Lastly, we show that under a mild condition on the knot group, the peripheral subgroup of a knotted surface is also a 0-concordance invariant.