We consider the monotone operator P, which maps Orlicz-Lorentz class into some ideal space Y=Y(R:+). Orlicz-Lorentz class is determined as the cone of Lebesgue-measurable functions on (Formula Presented) having the decreasing rearrangements that belong to weighted Orlicz space under some general assumptions concerning properties of functions (Formula Presented) and v. We prove the reduction theorems allowing reducing the estimates of the norm of operator (Formula Presented) to the estimates for its restriction on some cone of nonnegative step-functions in (Formula Presented). Application of these results to identical operator mapping (Formula Presented) into the weighted Lebesgue space (Formula Presented) gives the sharp description of the associate space for (Formula Presented). The main results of this paper were announced in, . They develop the results of our paper,  related to the case of N-functions. © 2017, Springer Nature Singapore Pte Ltd.