Generalization of results obtained from a paper that may be "false" I am a PhD student in mathematics, currently working on an article (article A) from two leading researchers in my field published in an 0.72 Impact journal.

I have an idea to generalize their results and publish my first article (paper B) but I have two problems:

In one of the proofs in paper A, the authors made a passage "crucial" which is not quite correct, according to several discussions with my supervisor .

What should I do with this situation? should I try to find another proof for the theorem and write it in a paper? 

In paper B, I will use theorems similar to those of Article A, but weakening the assumptions assumed in A.

Do you think there is any chance that this can be considered plagiarism?



Any other advice about what to do in this situation is welcome, and I would especially appreciate answers from mathematicians, as I have the impression that standard practices differ significantly from field to field.
What should I do [when a published result contains a mistake]? should I try to find another proof for the theorem and write it in a paper?


You cannot use their result as-is, because it is incorrect. If their result is sufficiently interesting, then you could perhaps publish a paper that corrects their result, otherwise, you could include a correction in your paper before using the corrected result.


  In the paper B, I will use theorems similar to those of Article A, but by weakening the assumptions assumed in A. Do you think there is any chance that this can be considered plagiarism?


No. 

You should make it clear that the theorems introduced in Article A are incorrect, and you should clearly motivate the introduction of your similar theorems that correct the originals. 

This isn't plagiarism. 



The OP clarified their plagiarism concerns as follows:


  my proofs will be similar to those used in paper A, except that I weakened the hypotheses. 


Before stating your proof, you could explain "The following proof builds upon a proof by Original Authors [Theorem X, Paper A]," you could go further and explain "novel aspects appear in the first, third and fourth paragraphs" or "novel aspects will be highlighted in the proof" (with suitable highlighting in the proof, e.g., "this aspect is new," possibly parenthesised).