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Hi. I have only able to find the definition of $Pic(R)$ for a commutative ring $R.$ Which is the isomorphism classes of projective $R$-modules of rank $1,$ and the product given by $[A][B]=[A\otimes_R B].$

How can i adapt this definition for the non-commutative case?

are there some books of articles related to this ? Thanks.

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You can take tensor invertible $R$-bimodules, but I mean, you must have a motivation to consider such things, and the right notion for you is the one that adapts best to your purposes. –  Fernando Muro Oct 17 '12 at 21:59
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up vote 4 down vote accepted

The definition I appreciate most comes from Morita theory: you consider unital (just for simplicity, something slightly weaker will also work) rings and the bimodules between them. Using the tensor product this makes almost a category: the tensor product of bimodules (over the ring in the middle) is associative up to a natural isomorphism and the rings themselves, viewed as bimodules over themselves with the mutliplication being the left/right module structures constitute the units, again up to a natural isomorphism. So either you prefer a bicategory setting, then you are done with what I said, or you prefer an honest category, then you have to pass to isomorphism classes of bimodules as morphisms between the rings. Of course, there are the usual set-theoretic issues that you should work in some universe etc.

But now the definition of the Picard groupoid is very simple: it is the groupoid of invertible arrows in this category. The Picard group of a single ring is then just the isotropy group at this ring of the big Picard group. In other words: the Picard group of $R$ is the group of invertible bimodules with respect to the tensor product as multiplication and the ring $R$ as unit.

It is then a famous theorem of Morita that characterizes the invertible bimodules $M$ between $R$ and $S$: there are various versions but a simple one is that they are finitely generated projective right modules over $R$ such that if you write $M = eR^n$ with an idempotent $e \in M_n(R)$ one has $ReR = R$, i.e. the two-sided ideal generated by the entries of the matrix $e$ is the whole ring $R$. Then the ring $S$ is given (via the left module structure) by the $R$-linear endomorphisms of $M$.

You can find this in many algebra textbooks like e.g. Lam's book on Modules and Rings, or Bass' book on Algebraic $K$-Theory. On my homepage you can also find some (very preliminary) lecture notes covering this stuff and other versions of Morita theory for rings with additional structures.

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Thanks a lot four your help. :) –  Hector Pinedo Dec 1 '12 at 0:22
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Amnon Yekutieli has studied the derived Picard group in a noncommutative setting. See Dualizing Complexes, Morita Equivalence and the Derived Picard Group of a Ring.

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