  Resource theory is a widely-applicable framework for analyzing the physical
resources required for given tasks, such as computation, communication, and
energy extraction. In this paper, we propose a general scheme for analyzing
resource theories based on resource destroying maps, which leave resource-free
states unchanged but erase the resource stored in all other states. We
introduce a group of general conditions that determine whether a quantum
operation exhibits typical resource-free properties in relation to a given
resource destroying map. Our theory reveals fundamental connections among basic
elements of resource theories, in particular, free states, free operations, and
resource measures. In particular, we define a class of simple resource measures
that can be calculated without optimization, and that are monotone
nonincreasing under operations that commute with the resource destroying map.
We apply our theory to the resources of coherence and quantum correlations
(e.g., discord), two prominent features of nonclassicality.
