  We prove that, for simple modules $M$ and $N$ over a quantum affine algebra,
their tensor product $M \otimes N$ has a simple head and a simple socle if $M
\otimes M$ is simple. A similar result is proved for the convolution product of
simple modules over quiver Hecke algebras.
  In the second version, the statement (1.11) (in the revised version) is
modified and its proof is given in Section 4.
