  We provide the first detailed study of the RR Lyrae period-luminosity (PL)
relation in the ugriz bandpasses of the Sloan Digital Sky Survey (SDSS) filter
system. We argue that tight, simple PL relations are not present in the SDSS
filters, except for the redder bandpasses i and (especially) z. However, for
all bandpasses, we show that, by incorporating terms involving a (fairly
reddening-independent) "pseudo-color" C_0 = (u-g)_0 - (g-r)_0, tight
(non-linear) relations do obtain. We provide theoretically calibrated such
relations in the present paper, which should be useful to derive precise
absolute magnitudes (hence distances) and intrinsic colors (hence reddening
values) to even {\em individual} field RR Lyrae stars. For applications to
cases where photometry in all five passbands may not be available, we also
provide simple (though less precise) average PL relations for the i and z
bandpasses, which read as follows:
  M_z = 0.839 - 1.295 log P + 0.211 log Z,
  M_i = 0.908 - 1.035 log P + 0.220 log Z.
  Similarly, simple period-color relations for (r-i)_0, (g-r)_0, and (u-z)_0
are also provided.
