  We define and study Hodge ideals associated to a coherent ideal sheaf J on a
smooth complex variety, via algebraic constructions based on the already
existing concept of Hodge ideals associated to Q-divisors. We also define the
generic minimal exponent of J, extending the standard invariant for
hypersurfaces. We relate it to Hodge ideals, and show that it is a root of the
Bernstein-Sato polynomial of J.
