I am preparing to write an article. For that, I got an easier proof of the already published result. My result is very simple to prove. Is it ethical to include this result in my article after citing the other author's work that we are improving their result? Kindly help. Thanks a lot.
New proofs are always good and of scientific value, and may be of significant use and interest in their own right. In some cases they are even worthy of a paper of their own. As an example, just recently on MathOverflow I saw Zhi-Wei sun mention that a result of his currently has 6 proofs from 6 different papers and sets of authors (including his own and himself).
It's better than ok, it's usually a Good Thing to do, since:
- It helps readers understand what the more complex result means, or what its significance is, by considering a simpler-to-perceive case.
- The simpler result is often powerful enough to be used itself - even if not for the specific use the original prover of the complex result had in mind.
- The simpler result can sometimes be established with a different, simpler proof, which may be of independent interest. Alternatively, if the proof is similar, the simpler result's proof helps understand the more complicated proof of the more complex result.
- Simpler results are often simpler to keep in mind.
Of course there's the question of your submission page limit - devoting space to a simpler results means condensing or removing other parts of the text, but that's something to decide based on the specifics of your article.
I agree with the two previous answers that publishing your simplified proof is good. Simplified proofs often lead to improved results; some of my own papers began with the idea that I can simplify a known proof but then developed into applications of the simplification, yielding new results. You might want to check whether your simplified proof can be easily extended to yield more than the original result.