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The phrase "extended the idea" is correct and usable in written English.
You can use it when discussing how a concept or notion has been elaborated or developed further in a conversation or text.
Example: "In her latest article, she extended the idea of sustainable living by incorporating new research on renewable energy sources."
Alternatives: "elaborated on the idea" or "expanded upon the idea.".
Exact(44)
Even if one's sensations portray oneself as spatially extended, the idea that one is non-extended (immaterial) is logically consistent with the presence of those sensations or (apparent) perceptual states.
But now Caspari has extended the idea.
So I have extended the idea of constructing an idealised world into constructing an idealised universe".
The Guardian extended the idea of openness during the last year to include publishing each day's newslist kept by editors.
The dual nature of light, however, was just beginning to gain scientific acceptance when de Broglie extended the idea of such a duality to matter.
In the Middle Ages the guilds of Europe and England extended the idea of mutual assistance to other circumstances of distress, such as illness.
Similar(15)
The program, which extended the ideas of the Marshall Plan beyond Europe, had been signed into law by President Harry S. Truman in 1951.
After that, in 1975, Benoit Mandelbrot extended the ideas of Julia.
Nieto et al. in [14] extended the ideas of [10] to prove the existence of solutions to some differential equations.
But plans are afoot to extend the idea into a number of new areas.
'Jams', that extend the idea of hack days beyond just software coding, were also proposed.
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Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com