  This paper discusses the evolution of the correlation between galaxy star
formation rates (SFRs) and stellar mass (M*) over the last ~10 Gyrs,
particularly focusing on its environmental dependence. We first present the
mid-infrared (MIR) properties of the H-alpha-selected galaxies in a rich
cluster Cl0939+4713 at z=0.4. With a MIR stacking analysis, we find that the
median SFR of H-alpha emitters is higher in higher-density environment at
z=0.4. We also find that star-forming galaxies in high-density environment tend
to have higher specific SFR (SSFR), although the trend is much less significant
compared to that of SFR. This increase of SSFR in high-density environment is
not visible when we consider the SFR derived from H-alpha alone, suggesting
that the dust attenuation in galaxies depends on environment; galaxies in
high-density environment tend to be dustier (by up to ~0.5 mag). We then
discuss the environmental dependence of the SFR-M* relation for star-forming
galaxies since z~2, by compiling our comparable, narrow-band selected, large
H-alpha emitter samples in both distant cluster environments (from
MAHALO-Subaru) and field environments (from HiZELS). We find that the SSFR of
H-alpha-selected galaxies (at the fixed mass of log(M*/Msun)=10) rapidly
evolves as (1+z)^3, but the SFR-M* relation is independent of the environment
since z~2, as far as we rely on the H-alpha-based SFRs (with M*-dependent
extinction correction). Even if we consider the possible environmental
variation in the dust attenuation, we conclude that the difference in the
SFR-M* relation between cluster and field star-forming galaxies is always small
(~<0.2dex level) at any time in the history of the Universe since z~2.
[abridged]
