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.
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