We have searched three hot molecular cores for submillimeter emission from the nucleic acid building-block pyrimidine. We obtain upper limits to the total pyrimidine (beam-averaged) column densities towards Sgr B2(N), Orion KL and W51 e1/e2 of 1.7E+14 cm^{-2}, 2.4E+14 cm^{-2} and 3.4E+14 cm^{-2}, respectively. The associated upper limits to the pyrimidine fractional abundances lie in the range (0.3-3)E-10. Implications of this result for interstellar organic chemistry, and for the prospects of detecting nitrogen heterocycles in general, are briefly discussed.
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