  A commutative integral domain is primary if and only if it is one-dimensional
and local. A domain is strongly primary if and only if it is local and each
nonzero principal ideal contains a power of the maximal ideal. Hence
one-dimensional local Mori domains are strongly primary. We prove among other
results, that if $R$ is a domain such that the conductor $(R:\widehat R)$
vanishes, then $\Lambda(R)$ is finite, that is, there exists a positive integer
$k$ such that each non-zero non-unit of $R$ is a product of at most $k$
irreducible elements. Using this result we obtain that every strongly primary
domain is locally tame, and that a domain $R$ is globally tame if and only if
$\Lambda(R)=\infty$. In particular, we answer Problem 38 in {P.-J. Cahen,
M.~Fontana, S.~Frisch, and S.~Glaz, Open problems in commutative ring theory,
Commutative Algebra, Springer 2014} in the affirmative. Many of our results are
formulated for monoids.
