Extensions of rings over 2-primal rings

Abstract

‎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.