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I have started doing a paper Applications of a New $K$-Theoretic Theorem to Soluble Group Rings by Kropholler which proves Kaplansky conjecture for soluble groups.

Now looking ahead in paper, I saw that it uses Grothendieck groups associated with all finitely generated right S-modules (denoted as $G_0(S)$), where S is a right noetherian ring. I have never had any introduction to them before but have heard about there usage in topology. I wont be looking much into them now, I only require them to understand this paper and its usage in them which is only associated with rings.

I am not looking in deeper study of these groups but only basic stuff and how they are associated with rings and modules.

Are there any online notes or some book recommendations, that can satisfy my requirements.

Bhaskar Vashishth
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    If you want something freely available, there's Weibel's K-Book. He's a pretty friendly writer and it seems like he covers the basics of $K_0(R)$ over ten pages or so in Chapter II, although I can't remember how often he needs things from Chapter I. – Hoot Jun 06 '15 at 23:59
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    Note that $G_0$ is slightly different than $K_0$. I would guess Weibel covers it, or you can pretty much figure it out from his treatment of $K_0$. – Kevin Carlson Jun 07 '15 at 00:41

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