I have no idea. と　I don't have an idea. どっちも正しいの？

ご質問を頂きました。お馴染み、George先生

の答えは、

The quick answer is yes...the more accurate answer is 'it depends'.

どっちも文法的には正しいですが、状況にもよる、ということですね。

"I have no idea" is a very common phrase that could also be said as "I don't have
any idea" which means you don't know the answer/what something means.

I have no idea.　やI don't have any idea. は「分からない」という意味。

"I don't have an idea" is a more situation-specific phrase that requires more information: "I don't have an idea for the school play."

I don't have an idea. は　for a school play など「なんの」アイデアか特定して使う、ということですね。

それでは、I have no +　物　のような使い方も出来るのでしょうか？

umbrella や　ticket で説明してもらいました。

As for 'I have no umbrella/ticket,' technically you can do it this way, though it isn't the most common way.

言えなくも無いけど、一般的ではない。

It really depends on the noun that follows it. Normally you would use "I don't have a ticket/umbrella", but the common usage for pride would be "I have no pride".

結局、名詞の性質によります。チケットやカサの場合は、

I don't have a ticket.
I don't have an umbrella.

pride　(誇り）の場合は、

I have no pride.

I would say that the best rule of thumb to use would be to use "I don't have a" and then a physical object, but use "I have no" for something intangible like pride, heart, feelings, etc.

I don't have a ～　は、物理的なモノに使い
Ihave no は、目に見えないもの
に使えばまず間違いない、ということです。