Abstract
In 2004 a counterexample was given for a 1965 result of R.J. Elliott claiming that discrete spectral synthesis holds on every Abelian group. Here we present a ring-theoretical approach to this problem, and show that some varieties fail to have spectral synthesis. In particular, we give a new proof for the result of the second author that spectral synthesis does not hold on Abelian groups with infinite torsion free rank.
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