## 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.

### 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

Zbl 1303.08004
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### 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
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