  We formulate notions of subadditivity and additivity of the Yang-Mills action
functional in noncommutative geometry. We identify a suitable hypothesis on
spectral triples which proves that the Yang-Mills functional is always
subadditive, as per expectation. The additivity property is much stronger in
the sense that it implies the subadditivity property. Under this hypothesis we
obtain a necessary and sufficient condition for the additivity of the
Yang-Mills functional. An instance of additivity is shown for the case of
noncommutative $n$-tori. We also investigate the behaviour of critical points
of the Yang-Mills functional under additivity. At the end we discuss few
examples involving compact spin manifolds, matrix algebras, noncommutative
$n$-torus and the quantum Heisenberg manifolds which validate our hypothesis.
