Polynomial rings over Jacobson-Hilbert rings.

*(English)*Zbl 0699.13011
Publ. Mat., Barc. 33, No. 1, 85-97 (1989); Addendum ibid. 34, No. 1, 223 (1990).

Starting point of the paper is the question whether the property of being SISI, i.e. every subdirect irreducible quotient of a commutative ring R \(with\quad 1\) is self-injective, extends to the polynomial ring R[x]. It is answered in the negative by a certain split-null extension (K,N) where K is an arbitrary field and N any vector space over K of not to small dimension. Furthermore the property of being “Monica” is investigated, meaning that every ideal I of R[x] for which R[x]/I is subdirectly irreducible (so-called COSI ideal) contains a monic polynomial \(\neq 0\). By means of a characterization of Jacobson-Hilbert rings (usually named Hilbert or Jacobson rings by Kaplansky and Bourbaki, respectively) by maximal ideals of R[x] it is shown in particular that a Noetherian ring is Monica iff it is Jacobson-Hilbert and that a Jacobson-Hilbert ring is Monica iff every quotient R[x]/I, I being a COSI ideal, is a local ring. Finally the properties SISI and Monica are characterized in the case of Morita rings, von Neumann rings and others.

Minor changes are given in the addendum.

Minor changes are given in the addendum.

Reviewer: G.Kowol

##### MSC:

13F20 | Polynomial rings and ideals; rings of integer-valued polynomials |

13C11 | Injective and flat modules and ideals in commutative rings |

13B25 | Polynomials over commutative rings |