# 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.

Copyright (c) 2019 Abdollah Alhevaz, Ebrahim Hashemi, Professor, Khadijeh Khalilnezhad, Phd. Student

This work is licensed under a Creative Commons Attribution 4.0 International License.

The authors retain all rights to the original work without any restrictions.

#### License for Published Contents

"Le Matematiche" published articlesa are distribuited with Creative Commons Attribution 4.0 International. You are free to copy, distribute and transmit the work, and to adapt the work. You must attribute the work in the manner specified by the author or licensor (but not in any way that suggests that they endorse you or your use of the work).

#### License for Metadata

"Le Matematiche" published articles metadata are dedicated to the public domain by waiving all publisher's rights to the work worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.

You can copy, modify, distribute and perform the work, even for commercial purposes, all without asking permission.

#### No Fee Charging

No fee is required to complete the submission/review/publishing process of authors paper.