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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.
Reviewer: G.Kowol

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
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