Extending UFDs to PIDs without adding units. (English) Zbl 1405.13037

The author answers the reviewer’s question related to convex subsets of generalized affine spaces (see [G. Czédli and A. B. Romanowska, Int. J. Algebra Comput. 23, No. 8, 1805–1835 (2013; Zbl 1303.08004)]): can the PID \(P \subseteq \mathbb{R}\) be chosen so that its notion of convexity is non-trivial? (This means that \(P\) properly contains \(\mathbb{Z}\) and has only \(\pm 1\) as units.)
The question is considered as a question of pure commutative ring theory, and is answered affirmatively. It is shown that for any UFD \(U\) there is a PID \(P\) containing \(U\) that has the same unit group as \(U\). The PID \(P\) can be chosen so that its field of fractions is a pure transcendental extension of the field of fractions of \(U\) with transcendence degree at most \(|U|\).
The PID which gives an answer for the question above is constructed as an extension of the UFD \(\mathbb{Z}[\pi]\). Some additional observations concerning extensions of UFD’s to PID’s are also explained.


13F15 Commutative rings defined by factorization properties (e.g., atomic, factorial, half-factorial)
13F10 Principal ideal rings
16U60 Units, groups of units (associative rings and algebras)
52A01 Axiomatic and generalized convexity


Zbl 1303.08004
Full Text: DOI Euclid


[1] M.F. Atiyah and I.G. Macdonald, Introduction to commutative algebra, Addison-Wesley Publishing Co., Reading, MA, 1969. · Zbl 0175.03601
[2] G. Czédli and A.B. Romanowska, Generalized convexity and closure conditions, Internat. J. Algebra Comp. 23 (2013), 1805–1835. · Zbl 1303.08004
[3] D. Dummit and R. Foote, Abstract algebra, Third edition, John Wiley & Sons, Inc., Hoboken, NJ, 2004. · Zbl 1037.00003
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.