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

Kearnes, Keith A.

J. Commut. Algebra 10, No. 3, 347-358 (2018).

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.

Reviewer: Anna Romanowska (Warsaw)

MSC:

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

Keywords:

UFD; PID; Euclidean domain; unit group; affine space; convex set

Citations:

Zbl 1303.08004

References:

[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