Extensions of rings over 2-primal rings

‎For a set of endomorphisms $\Sigma‎ :‎= \{\sigma _1,\ldots‎ , ‎\sigma _n\}$ and derivations $\Delta‎ :‎= \{\delta _1,\ldots‎ , ‎\delta _n\}$‎, ‎we first introduce $\Sigma$-compatible ideals which are a generalization of $\Sigma$-rigid ideals and study the connections of the prime radical and the upper nil radical of $R$ with the prime radical and the upper nil radical of the skew PBW extension‎. ‎Let $A = R \left\langle x_1‎, ‎\ldots‎ , ‎x_n; \Sigma‎, ‎\Delta‎ ‎\right\rangle $ be an injective skew PBW extension of an‎ ‎$(\Sigma,\Delta)$-compatible ring $R$‎. ‎(i) It is shown that if‎ ‎$R$ is a (semi)prime ring‎, ‎then $A $ is a (semi)prime ring‎. ‎(ii)‎ ‎If $R$ is a completely (semi)prime ring‎, ‎then $A $ is a‎ ‎completely (semi)prime ring‎. ‎(iii) If $R$ is a strongly‎ ‎(semi)prime ring‎, ‎then $A $ is a strongly (semi)prime ring‎. ‎Also‎, ‎we prove that $R$ is $2$-primal if and only if the‎ ‎injective skew PBW extension $A $ is $2$-primal if and only if‎ ‎$nil(R) = nil_{*}(R; \Sigma \cup \Delta)$ if and only if $‎ ‎nil(R)\left\langle x_1‎, ‎\ldots‎ , ‎x_n; \Sigma,\Delta\right\rangle‎ ‎= nil_*(A)$ if and only if every minimal $(\Sigma,\Delta)$-prime‎ ‎ideal of $R$ is completely prime‎