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問題ない vs 問題じゃない
From what I understand the negative form of a noun is created with じゃない and not ない however my textbook uses 問題ない in an example (translated as "no problem").
Looking elsewhere on the Internet it looks like 問題ない seems to be acceptable usage but I do not understand why within the rules of the language.
問題ない does not mean 'It's not a problem', which is what 問題じゃない would mean. You may be aware that ない is the negative form of ある, 'there is', which means that 問題ない means 'There is no problem.' Note that you can also say 問題はない.
問題ない is not negating the noun (that would be 問題じゃない or 問題ではない, as you correctly noticed). Rather, it is saying 問題がない (or 問題はない), literally "there isn't a problem".
I think you can safely treat 問題ない as a set phrase, rather than an ellipsis (omission) of が or は. (For example, I don't think 問題ない is particularly informal, although omissions usually are.)
There is at least one other similar phrase, 異常なし ("No concerns"). You can read about なし (an older variant of ない) and ない in detail here: What does 「なし」in 「問題なし」 mean?
