Ring structure of SUSY * product and 1/2 SUSY Wess-Zumino model

There are two types of non(anti)commutative deformation of D=4, N=1 supersymmetric field theories and D=2, N=2 theories. One is based on the nonsupersymmetric star product and the other is based on the supersymmetric star product. These deformations cause partial breaking of supersymmetry in general. In case of supersymmetric star product, the chirality is broken by the effect of the supersymmetric star product, then it is not clear that Lagrangian or observables including F-terms preserve part of supersymmetry. In this Letter, we investigate the ring structure whose product is defined by the supersymmetric star product. We find the ring whose elements correspond to 1/2 SUSY F-terms. Using this, the 1/2 SUSY invariance of the Wess-Zumino model is shown easily and directly