Ribenboim, Paulo The ascending chain condition for real ideals. (English) Zbl 0611.13002 Manuscr. Math. 57, 109-124 (1986). Let R(S) denote the smallest real ideal containing a set S. In this paper the author imitates the standard proofs from commutative algebra to show that a ring A satisfies the acc for real ideals if and only if every real prime is R(S) for a finite set S (in fact a one-element set) and that A[X] satisfies the acc for real ideals if and only if A[X] does. He shows by counterexample that this is not true for A[[X]]. Reviewer: R.O.Robson MSC: 13A15 Ideals and multiplicative ideal theory in commutative rings 13E15 Commutative rings and modules of finite generation or presentation; number of generators 14Pxx Real algebraic and real-analytic geometry 13F20 Polynomial rings and ideals; rings of integer-valued polynomials Keywords:Hilbert basis theorem; real prime; acc for real ideals PDFBibTeX XMLCite \textit{P. Ribenboim}, Manuscr. Math. 57, 109--124 (1986; Zbl 0611.13002) Full Text: DOI EuDML References: [1] LAM, T.Y.: Introduction to real algebra. Rocky Mtn. J.14, 767-814 (1984) · Zbl 0577.14016 · doi:10.1216/RMJ-1984-14-4-767 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.