A gerund is a verbal word or phrase which functions as a noun or noun phrase.
The verb improve can be used strictly as a verb, as in the following sentence:
I improve my performance with each successive attempt.
It can also be used as a gerund and function as a noun. For example,
Improving is my goal with each and every performance.
Here, improving is the subject of the sentence.
Of your first group of three sentences, each one has its minor flaws:
"I worked on the improvement of the article." Unless someone asked the question of, say, a group of people, "Who worked on the improvement of the article? Show of hands," you could probably drop the word the. A better version would then be "I worked on improving the article."
"I worked on the article improving." You could make this sentence correct by adding a comma and some words after the word improving. For example, "I worked on the article, improving its style and point of view." Otherwise, you'd need to reword the sentence for better clarity. Some examples:
I worked on the article's improvement.
Working on improving the article was my pleasure.
Improving the article was my work [or contribution, or task, or responsibility, and so on].
My work on improving the article was exhausting.
"I worked on the improving of the article." You could say
"I worked on improving the article" [no the is needed, and the gerund becomes part of a prepositional phrase], or
" I worked on the improvement of the article." Here the word the is needed.
As for your second group of sentences, the first one is fine, but the second one,
"Investing idea was published,"
is not, since the words investing idea need an identifier/mofifier such as the, an, his, her, their, my professor's, my uncle's, my sister's, and so on.
By the way, you would not detract from the sentence's meaning by substituting the word investment for investing, as in
My investment idea was published. Investment here functions as a modifier of the word idea.
The same could be said for the following variation:
My ideas on investing were published. Investing functions here as the (noun) object of the preposition on.