Why isn't there an article in this sentence? : "Mine computer is broken."

The sentence is incorrect.  It should be: "**My** *computer is broken.*"

"My computer is broken" (Not mine :) ) If it wasn't your computer, you'd say "A computer is broken / The computer is broken / Some computers are broken? But since you're using a possessive pronoun ("my"), you don't need any of the above. If you WANTED to, you could try to include both "The computer which is mine is broken", but it's longer and sounds a little unnatural.

That sentence is grammatically wrong in modern English. It should be "My computer," not "Mine computer."

The correct sentence is, "my computer is broken." But to answer your actual question: Articles (a/an/the) are part of a larger group called determiners. Determiners include articles, demonstrations (that, these, this, those), quantifiers (many, few, some, no), numbers (one, five, zero) AND possessives (my, your, their, his)! You need a determiner at the start of this sentence, but you don't need more than one. So: A computer is broken. My computer is broken. This computer is broken. One computer is broken.

Think of the article as trying to explain WHICH thing you're talking about, in the sentence "my computer is broken," you've already established which computer: your computer. If you had two computers, one upstairs and one downstairs, then you might say "the upstairs computer is broken" or "my upstairs computer is broken." In both cases you're telling the listener which computer you mean, but you use a different method in each sentence. One uses the definite article, and one doesn't. You'd be more likely to use "the" if you're speaking to someone who lives with you, since they already know it's yours and that you have an upstairs computer, whereas you'd be more likely to use "my" if you're speaking to someone who didn't know you had an upstairs computer.

English doesn't add articles with possessive pronouns. So "my computer" not "the my computer"

Correct way to say it is: my computer is broken Although your sentence might be an old fashioned way of saying this

Everyone has answered your core question, but I have another item to mention: I get what you meant, but I’ve never heard rock bottom (“from the rock bottom up”) used that way. To me it sounds a bit off, but I would appreciate the perspective of other native speakers.

It’s very archaic

lmao this feels like a fake German sentence, “Mein komputer ist brokenen (sic)”.

So as others have said, it should be "My computer is broken." To understand why there is no article, you have to understand what an article /does/. It fills a role in English called "determiner". Most noun phrases in English are required to have a determiner. But articles aren't the only things that can act as determiners - there are lots of others. In this case, the possessive particle "my" is filling the determiner role.

This looks like a parody of archaic speech. We used to use "mine" similarly to "my" hundreds of years ago (although I think it was only before a vowel, like with "a" and "an"). Obviously computers weren't invented yet, so "mine computer" sounds funny. 🙃

My friends and I do this as an obvious poor Shakespearean impression and Ive seen other people do this as well. Is it grammatically correct? No, but it's used for comedic effect.

Articles are used for specificity. Which computer? A computer. or the computer? In this case, my computer. The article isn't necessary because the possessive pronoun supplies the specificity.

out of curiosity whats your native language?