While studying I came across this short 会話{かいわ}：

> A: この本をもらってもいいですか？  
>  B: ええ、いいですよ。どうぞ。

A was translated as: "Can I keep this book?" Now, I've learned that もらう means
to receive (something). I would have thought that another word would be used
for "keeping" or "taking ownership" of another object. Am I wrong?
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verbs
Can もらう mean to keep?
Though もらう does have a similar meaning to "receive", that doesn't quite tell
the whole story. Used as a normal verb (not an auxiliary) as in your example,
the most [first two definitions] are as follows:

>   1. **贈【おく】られたり頼【たの】んだりして受【う】け取【と】り、自分【じぶん】のものとする。**
> 「金【かね】を―・う」「便【とよ】りを―・う」「賞【しょう】を―・う」「元気【げんき】を―・う」「勇気【ゆうき】を―・う」
>
>   2. **頼【たの】んで手【て】に入【い】れる。得【え】る。** 「許可【きょか】を―・う」
>
>

The most important part of these definitions is (in my opinion) 「自分のものとする」 (to
make something one's own). So although the meaning of "receive" is certainly
included in there, it also overlaps with "take", "accept", and "acquire".

In this particular instance, the question 「もらっていいですか」 is asking if it is
really to accept the book, i.e. to make it A's own.
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もらっていい（ですか） is a [pretty common way of asking for
something](http://eow.alc.co.jp/search?q=Can%20I%20keep%20this?), and I would
translate it either as "can I keep this" or "can I have this?" As you said,
もらう is receive, so it's not hard to see how this actually works out.

There are several ways to ask for something, each with its own nuances, but
using もらっていい will generally mean an explicit change of ownership. If you want
to make it more polite, いただけますか will also have the same sort of connotation.

On the other hand, if you want to ask for something _without_ keeping it, you
can use a variety of other verbs directly relating to what you want to do,
like その本を見ていいですか？ or 使っていい？ If you want to keep it general, you can say, for
example, この本を貸してくれますか？ or 借りていい？, meaning "would you lend" and "can I borrow"
respectively.
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You're not wrong about もらう but you need to realize もらっていい is not the same
thing as もらう.

彼は本をもらった means he received the book.

この本をもらっていい is a grammar pattern for questions where Vてform + いい which asks
whether the state of Vて is acceptable.

In other words, "is it okay/alright if I receive the book?"

Other samples of the same pattern:

> 質問聞いていい?

May I ask a question?

> 自分の番始めていい?

Can I start my turn?

> 焼いていい?

Can I fry/cook/bake it?
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