In my opinion: "a Kate's idea" is not a grammatically valid noun-phrase because the indefinite article ("a", "an") can't precede the possessive form of a proper noun Generally speaking a noun phrase commencing with the possessive form of a proper noun is equivalent to a noun phrase starting with the definite article ("the") in the sense that you are (to an extent) assuming that the speaker can identify the thing with reference to the noun-phrase. The correct singular form when you are *not* making that assumption is: indefinite article ("a" or "an") + noun + "of" + possessive form of proper noun e.g. "a friend of Kate's" For the plural: noun (plural) + "of" + possessive form of proper noun e.g. friends of Kate's A similar pattern happen with pronouns: "my book" has a similar quality of definiteness to "the book" "a book of mine" has a similar quality of indefiniteness to "a book"

Because the meaning of "Kate's idea" is :"the idea of Kate" Then as if you are saying: "That's a good the idea of Kate" And this doesn't fit because after the adjective good musn't be a "the"

Kate's idea is a great one. I like Kate's idea. That's a great idea. Kate thought of it. We owe this good idea to Kate. That's Kate's idea. I think it's a good one. From Kate, I heard this great idea. I don't know if it is correct or not, but everyone says: That's a good idea of Kate's. It's probably more correct to say: That's a good idea of Kate. Your question of why one can't say "that's a good Kate's idea" is a good one. My answer is "I don't know, but you can't". You can't put "Kate's" between a determiner and a noun. All of these are illegal: the Kate's idea a Kate's idea this Kate's idea that Kate's idea some Kate's idea any Kate's idea, etc. .... Instead, you have to say "the idea of Kate", "this idea of Kate", etc...

This question really got me thinking. I think the answer is something like: You can't mix a possesive and other adjectives using this construction. That's John's dog. That's a good dog. (CORRECT) That's a good John's dog. (INCORRECT) Part of the reason may be that the article applies to both, and in that case it is incorrect. That's a good idea. (correct) That's a Kate's idea (incorrect).

Hello! Not sure what you want to say. If you are saying Kate had a good idea to Kate you should say: "Kate, that's a good idea!" Or "What a good idea, Kate!" If you are talking to someone else about Kate's idea you can say: "Kate had a good idea" or "That's Kate's good idea". You can't say "That's a good Kate's idea" for the following reasons: First, "Kate" is in the middle of your sentence so we don't know her role in the action. The expression "That's a good idea" has a subject (That), verb ('s=is) and an object (a good idea). But what about Kate? In your original sentence you have put "Kate+'s+idea". indicating that Kate is part of the object of the sentence. But "good" and "idea" have to be together because they are an adjective and a noun that interact with each other: it's not Kate that's good, but her idea. To express the sentence correctly you should write: subject, verb, person, complement. That (subject) 's/is (verb) Kate's (person) good idea (complement). You can also write a different sentence with this order: subject, verb, complement, person. That (subject) 's/is (verb) a good idea (complement), Kate (person). So you can write the sentence in two ways: That's a good idea, Kate! (if you are talking to Kate) and "That's Kate's good idea" (talking to someone else). Hope to helps!