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**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.

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 |

### Citations:

Zbl 1303.08004
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\textit{K. A. Kearnes}, J. Commut. Algebra 10, No. 3, 347--358 (2018; Zbl 1405.13037)

### 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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