In this paper, we prove the following Riesz spaces’ version of the Korovkin theorem. Let E and F be two Archimedean Riesz spaces with F uniformly complete, let W be a nonempty subset of $$E^{+}$$ , and let $$(T_{n})$$ be a given sequence of (r-u)-continuous elements of $$\mathcal {L(}E,F)$$ , such that $$\left| T_{n}-T_{m}\right| x=\left| (T_{n}-T_{m})x\right| \mathcal {\ }$$ for all $$x\in E^{+},$$ $$m,n\ge n_{0}$$ (for a given $$n_{0}\in \mathbb {N} )$$ . If the sequence $$(T_{n}x)_{n}$$ $$(r-u)$$ -converges for every $$x\in W$$ , then $$(T_{n})$$ $$(r-u)$$ -converges also pointwise on the ideal $$E_{W}$$ , generated by W, to a linear operator $$S_{0}:E_{W}\rightarrow F$$ . We also prove a similar Korovkin-type theorem for nets of operators. Some applications for f-algebras and orthomorphisms are presented.