  We define a Rasmussen $s$-invariant over the coefficient ring of the
integers, and show how it is related to the $s$-invariants defined over a
field. A lower bound for the slice genus of a knot arising from it is obtained,
and we give examples of knots for which this lower bound is better than all
lower bounds coming from the $s$-invariants over fields. We also compare it to
the Lipshitz-Sarkar refinement related to the first Steenrod square.
