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

Wessel, Daniel

J. Commut. Algebra 13, No. 4, 583-588 (2021).

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.

Reviewer: Marco Benini (Buccinasco)

MSC:

13B25	Polynomials over commutative rings
03F65	Other constructive mathematics

Keywords:

connected rings; constructive algebra; Krull’s lemma; polynomials

Citations:

Zbl 1080.13002

References:

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