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**A note on connected reduced rings.**
*(English)*
Zbl 1481.13016

This short note illustrates a constructive proof of a characterisation of reduced and connected rings, whereas the cassical proof makes an apparently fundamental use of prime ideals and Krull’s Lemma.

Avoiding these constructively non-admissible tools has been achieved by using instead radical ideals and their properties, which are constructively acceptable.

The most interesting part of the paper is Section 3, which provides a precise yet intuitive insight on the reasons why the approach works and thus how it can be applied on other problems.

For a reader who does not have already in mind the specific literature, the article could be hard to read since it heavily relies on the cited works, in particular [S. McAdam and R. G. Swan, Lect. Notes Pure Appl. Math. 236, 411–424 (2004; Zbl 1080.13002)]. Hence, it is recommended to read the paper together or after [loc. cit.], so that the neat and clean exposition of the note is appropriately complemented by the background in the latter work.

Avoiding these constructively non-admissible tools has been achieved by using instead radical ideals and their properties, which are constructively acceptable.

The most interesting part of the paper is Section 3, which provides a precise yet intuitive insight on the reasons why the approach works and thus how it can be applied on other problems.

For a reader who does not have already in mind the specific literature, the article could be hard to read since it heavily relies on the cited works, in particular [S. McAdam and R. G. Swan, Lect. Notes Pure Appl. Math. 236, 411–424 (2004; Zbl 1080.13002)]. Hence, it is recommended to read the paper together or after [loc. cit.], so that the neat and clean exposition of the note is appropriately complemented by the background in the latter work.

Reviewer: Marco Benini (Buccinasco)

### Citations:

Zbl 1080.13002
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\textit{D. Wessel}, J. Commut. Algebra 13, No. 4, 583--588 (2021; Zbl 1481.13016)

### References:

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[7] | D. Rinaldi and P. Schuster, “A universal Krull-Lindenbaum theorem”, J. Pure Appl. Algebra 220:9 (2016), 3207-3232. · Zbl 1420.03128 |

[8] | D. Rinaldi, P. Schuster, and D. Wessel, “Eliminating disjunctions by disjunction elimination”, Bull. Symb. Log. 23:2 (2017), 181-200. · Zbl 1455.03074 |

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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.