An interesting branch of recent research in commutative algebra is devoted to understanding the local rings over which every totally reflexive module is free. Let \((R,\mathcal{M},k)\) be a local ring with canonical module \(\omega\). It is well-known that if \(k\) is a direct summand of some syzygy of \(\omega\), then every totally reflexive \(R\)-module is free. The main result in this paper asserts the following: Assume that \(R\) is artinian and almost Gorenstein in the sense of C. Huneke and A. Vraciu [Pac. J. Math. 225, No. 1, 85–102 (2006; Zbl 1148.13005)], but not Gorenstein. Write \(R=S/J\), where \(S\) is an artinian Gorenstein ring and let \(c\) be the \(k\)-vector space dimension of \((J :_S \mathcal{M})/(\mathcal{M}J :_S \mathcal{M})\). Then \(k^c\) is a direct summand of the second syzygy of \(\omega\). As a consequence, the authors conclude that over a Teter ring, every totally reflexive module is free.
For the entire collection see [Zbl 1226.13001].