  We construct an algebraic-cycle based model for the motivic cohomology on the
category of schemes of finite type over a field, where schemes may admit
arbitrary singularities and may be non-reduced.
  We show that our theory is functorial on the category, that it detects
nilpotence, and that its restriction to the subcategory of smooth schemes
agrees with the pre-existing motivic cohomology theory, which is the higher
Chow theory of S. Bloch (Adv. Math., 1986). A few structures and applications
are discussed.
