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 [\textit{S. McAdam} and \textit{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.