Compatible ideals and radicals of Ore extensions.

*(English)*Zbl 1106.16029Rings considered in this paper are associative with identity. For a given ring \(R\), \(R[x;\alpha,\delta]\) denotes the Ore extension of \(R\) and \(R[\![x;\alpha]\!]\) stands for the skew power series ring, where \(\alpha\) is an endomorphism and \(\delta\) an \(\alpha\)-derivation of \(R\), that is, \(\delta\) is an additive map such that \(\delta(ab)=\delta(a)b+\alpha(a)\delta(b)\) for \(a,b\in R\). \(P(R)\) and \(N_r(R)\) denote the prime radical and the upper nil radical of \(R\), respectively. A ring \(R\) is called strongly prime if \(R\) is prime with no nonzero nil ideals. An ideal \(P\) of \(R\) is strongly prime if \(R/P\) is a strongly prime ring. All (strongly) prime ideals are taken to be proper. An ideal \(P\) of a ring \(R\) is called minimal (strongly) prime if \(P\) is minimal among (strongly) prime ideals of \(R\). An ideal \(P\) of \(R\) is completely prime (completely semiprime) if \(ab\in P\) implies \(a\in P\) or \(b\in P\) (\(a^2\in P\) implies \(a\in P\)) for \(a,b\in R\). For an endomorphism \(\alpha\) of a ring \(R\), an \(\alpha\)-ideal \(I\) is called an \(\alpha\)-rigid ideal if \(a\alpha(a)\in I\) implies \(a\in I\) for \(a\in R\). The authors call an ideal \(I\) of \(R\) an \(\alpha\)-compatible ideal of \(R\) if \(ab\in I\) if and only if \(a\alpha(a)\in I\) for each \(a,b\in R\). They call \(I\) a \(\delta\)-compatible ideal of \(R\) if \(ab\in I\) implies \(a\delta(a)\in I\) for each \(a,b\in R\). \(I\) is said to be an \((\alpha,\delta)\)-compatible ideal of \(R\) if \(I\) is both, \(\alpha\)-compatible and \(\delta\)-compatible.

The authors give several necessary and sufficient conditions which assure that \(I\) is an \(\alpha\)-rigid ideal (\(\delta\)-ideal) of \(R\). In particular, they show that \(I\) is an \(\alpha\)-rigid ideal of \(R\) if and only if \(I\) is \(\alpha\)-compatible and completely semiprime. Also, they give examples of \(\alpha\)-compatible ideals which are not \(\alpha\)-rigid. They study connections between \((\alpha,\delta)\)-compatible ideals of \(R\) and related ideals of \(R[x;\alpha,\delta]\) and \(R[\![x;\alpha]\!]\). Moreover, they investigate the relationship of \(P(R)\) and \(N_r(R)\) of \(R\) with the prime radical and the upper nil radical of \(R[x;\alpha,\delta]\) and \(R[\![x;\alpha]\!]\). They show, in particular, that if \(I\) is a (semi) prime \((\alpha,\delta)\)-compatible ideal of \(R\), then \(I[x;\alpha,\delta]\) is a (semi) prime ideal of \(R[x;\alpha,\delta]\). Thus, if each mnimal prime ideal of \(R\) is \((\alpha,\delta)\)-compatible, then \(P(R[x;\alpha,\delta])\subseteq P(R)[x;\alpha,\delta]\). They prove that if \(P\) is a completely (semi) prime \((\alpha,\delta) \)-compatible ideal of \(R\), then \(P[x;\alpha,\delta]\) is a completely (semi) prime ideal of \(R[x;\alpha,\delta]\). From this the authors deduce the result of Hong, Kwak and Rizvi which states that if \(P(R)\) is an \(\alpha\)-rigid \(\delta\)-ideal of \(R\), then \(P(R[x;\alpha,\delta])\subseteq P(R)[x;\alpha,\delta]\). The authors also show that if \(P\) is a strongly prime \((\alpha,\delta)\)-compatible ideal of \(R\), then \(P[x;\alpha,\delta]\) is a strongly prime ideal of \(R[x;\alpha,\delta]\). Consequently, if each minimal strongly prime ideal of \(R\) is \((\alpha,\delta) \)-compatible, then \(N_r(R[x;\alpha,\delta])\subseteq N_r(R)[x;\alpha,\delta]\). This allows them to obtain the result of Hong, Kwak and Rizvi which states that if \(N_r(R)\) is an \(\alpha\)-rigid \(\delta\)-ideal of \(R\), then \(N_r(R[x;\alpha,\delta])\subseteq N_r(R)[x;\alpha,\delta]\). Parallel results to the above are also proved for \(R[\![x;\alpha]\!]\).

The authors give several necessary and sufficient conditions which assure that \(I\) is an \(\alpha\)-rigid ideal (\(\delta\)-ideal) of \(R\). In particular, they show that \(I\) is an \(\alpha\)-rigid ideal of \(R\) if and only if \(I\) is \(\alpha\)-compatible and completely semiprime. Also, they give examples of \(\alpha\)-compatible ideals which are not \(\alpha\)-rigid. They study connections between \((\alpha,\delta)\)-compatible ideals of \(R\) and related ideals of \(R[x;\alpha,\delta]\) and \(R[\![x;\alpha]\!]\). Moreover, they investigate the relationship of \(P(R)\) and \(N_r(R)\) of \(R\) with the prime radical and the upper nil radical of \(R[x;\alpha,\delta]\) and \(R[\![x;\alpha]\!]\). They show, in particular, that if \(I\) is a (semi) prime \((\alpha,\delta)\)-compatible ideal of \(R\), then \(I[x;\alpha,\delta]\) is a (semi) prime ideal of \(R[x;\alpha,\delta]\). Thus, if each mnimal prime ideal of \(R\) is \((\alpha,\delta)\)-compatible, then \(P(R[x;\alpha,\delta])\subseteq P(R)[x;\alpha,\delta]\). They prove that if \(P\) is a completely (semi) prime \((\alpha,\delta) \)-compatible ideal of \(R\), then \(P[x;\alpha,\delta]\) is a completely (semi) prime ideal of \(R[x;\alpha,\delta]\). From this the authors deduce the result of Hong, Kwak and Rizvi which states that if \(P(R)\) is an \(\alpha\)-rigid \(\delta\)-ideal of \(R\), then \(P(R[x;\alpha,\delta])\subseteq P(R)[x;\alpha,\delta]\). The authors also show that if \(P\) is a strongly prime \((\alpha,\delta)\)-compatible ideal of \(R\), then \(P[x;\alpha,\delta]\) is a strongly prime ideal of \(R[x;\alpha,\delta]\). Consequently, if each minimal strongly prime ideal of \(R\) is \((\alpha,\delta) \)-compatible, then \(N_r(R[x;\alpha,\delta])\subseteq N_r(R)[x;\alpha,\delta]\). This allows them to obtain the result of Hong, Kwak and Rizvi which states that if \(N_r(R)\) is an \(\alpha\)-rigid \(\delta\)-ideal of \(R\), then \(N_r(R[x;\alpha,\delta])\subseteq N_r(R)[x;\alpha,\delta]\). Parallel results to the above are also proved for \(R[\![x;\alpha]\!]\).

Reviewer: Halina France-Jackson (Port Elizabeth)

##### MSC:

16S36 | Ordinary and skew polynomial rings and semigroup rings |

16D25 | Ideals in associative algebras |

16N80 | General radicals and associative rings |

16N60 | Prime and semiprime associative rings |

16W25 | Derivations, actions of Lie algebras |

16W20 | Automorphisms and endomorphisms |

16N40 | Nil and nilpotent radicals, sets, ideals, associative rings |

16W60 | Valuations, completions, formal power series and related constructions (associative rings and algebras) |