We have searched for the production of a selectron and a squark in $e^+p$ collisions at a center-of-mass energy of 300 GeV using the ZEUS detector at HERA. The selectron and squark are sought in the direct decay into the lightest neutralino in the framework of supersymmetric extensions to the Standard Model which conserve R-parity. No evidence for the production of supersymmetric particles has been found in a data sample corresponding to 46.6~pb$^{-1}$ of integrated luminosity. We express upper limits on the product of the cross section times the decay branching ratios as excluded regions in the parameter space of the Minimal Supersymmetric Standard Model.
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