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.
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.
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.
