We construct the supercurrent and a supersymmetric current which satisfies the Adler-Bardeen theorem in supersymmetric Yang-Miils theory coupled to non-self-interacting chiral matter. Using the formulatiori recently developed by Grisaru, Milewski, and Zanon, supersymmetry and gauge invariance are maintained with supersymmetric background-field theory and regularization by dimensional reduction. We verify the finiteness of the supercurrent to one loop, and the Adler-Bardeen theorem to two loops by explicit calculations in the minimal-subtraction scheme. We then demonstrate the subtraction-scheme independence of the one-loop Adler-Bardeen anomaly and prove the existence of a subtraction scheme in which the Adler-Bardeen theorem is satisfied to all orders in perturbation theory. Soon after, ones' "solved" the consistency problem in an apparently simple way which also provided an allorders derivation of the / 3 function. His arguments were the following. The component G of the anomaly superfield which contains the axial anomaly has the form If the AB theorem is assumed, the anomalous divergence of the axial-vector current is equal to its one-loop value:Since it is the first component of the supercurrent, the anomalous divergence of j: should also be related to the /3 function and the anomaly superfield component G by Given these assumptions, not only are Eq. (1.2), which represents the AB theorem, and Eq. (1.31, the supercurrent equation, consistent, but they can also be solved for the all-orders /3 function. Amazingly, this procedure gives the correct two-loop value for the /3 function in pure supersymmetric Yang-Mills theory. Jones and

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