Prime ideals in skew and \(q\)-skew polynomial rings.

*(English)*Zbl 0814.16026
Mem. Am. Math. Soc. 521, 106 p. (1994).

Until recently, most research on skew polynomial rings \(R[y;\tau,\delta]\), where \(\tau\) is an automorphism of the ring \(R\) and \(\delta\) is a \(\tau\)-derivation of \(R\) has been concentrated in cases where either \(\tau\) or \(\delta\) is trivial, that is \(\tau\) is the identity or \(\delta = 0\). This has now changed because of the emergence of many interesting algebras, including some quantized enveloping algebras and coordinate rings of quantum groups, which are iterated skew polynomial rings involving, in at least one step, an extension of the above form with both \(\tau\) and \(\delta\) non-trivial.

The memoir under review is a comprehensive study of prime ideals in \(R[y;\tau,\delta]\) and is likely to become a standard reference, both for its results on prime ideals and for some of the basic computations for rings of this kind. It is written at three levels of generality. Several sections are written under only the assumption of a Noetherian coefficient ring. The results are then sharpened in the case where the derivation is \(q\)-skew, that is \(\delta\tau = q\tau \delta\) for some central element \(q\) of \(R\) such that \(\tau(q) = q\) and \(\delta(q) = 0\). Finally, there are two sections applying the results to particular examples.

One significant new notion with potential for use in other contexts is that of a prime ideal \(P\) of a ring \(S\) ‘lying directly over’ a prime ideal \(Q\) of a subring \(R\). This means that \(P\) lies over \(Q\) in the usual sense and that \(Q\) is an annihilator prime of the right \(R\)- module \(S/P\). For the rings considered here, this notion is sharper than the usual one of lying over.

The memoir is very clearly written and is liberally illustrated with helpful and interesting examples.

The memoir under review is a comprehensive study of prime ideals in \(R[y;\tau,\delta]\) and is likely to become a standard reference, both for its results on prime ideals and for some of the basic computations for rings of this kind. It is written at three levels of generality. Several sections are written under only the assumption of a Noetherian coefficient ring. The results are then sharpened in the case where the derivation is \(q\)-skew, that is \(\delta\tau = q\tau \delta\) for some central element \(q\) of \(R\) such that \(\tau(q) = q\) and \(\delta(q) = 0\). Finally, there are two sections applying the results to particular examples.

One significant new notion with potential for use in other contexts is that of a prime ideal \(P\) of a ring \(S\) ‘lying directly over’ a prime ideal \(Q\) of a subring \(R\). This means that \(P\) lies over \(Q\) in the usual sense and that \(Q\) is an annihilator prime of the right \(R\)- module \(S/P\). For the rings considered here, this notion is sharper than the usual one of lying over.

The memoir is very clearly written and is liberally illustrated with helpful and interesting examples.

Reviewer: D.A.Jordan (Sheffield)

##### MSC:

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

16D25 | Ideals in associative algebras |

16-02 | Research exposition (monographs, survey articles) pertaining to associative rings and algebras |

16P40 | Noetherian rings and modules (associative rings and algebras) |

16W25 | Derivations, actions of Lie algebras |