A 2-knot is a 2-sphere smoothly embedded in the 4-sphere. In 1977 Melvin defined an equivalence relation on 2-knots called 0-concordance, and proved that 0-concordant 2-knots have diffeomorphic Gluck twists. This prompted the question, are all 2-knots 0-concordant to the unknot? Recently this was shown to be false by Sunukjian and Dai-Miller using techniques from Heegaard Floer homology applied to special Seifert 3-manifolds for the 2-knots. In this talk we will present another proof using Alexander ideals. The main result is that the Alexander ideal induces a homomorphism from the 0-concordance monoid of 2-knots to the ideal class monoid of the ring of integral Laurent polynomials. A corollary is that any 2-knot with nonprincipal Alexander ideal has no inverse in the 0-concordance monoid.

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