The depth of a ring or module is one of the most basic invariants in commutative ring theory.
Q1: Is there also a powerful notion of depth for non-commutative rings ?
By a search in mathscinet, I only found the papers [BT], [R]. In the latter the $I$-depth of a module is defined (in analogue to the comm. case) as the minimal $i$ such that $Ext_R^i(R/I,M)\neq 0$ ($R$ a ring, $I$ a left ideal in $R$ and $M$ a left $R$-module). But there are no applications of the depth in the paper. In particular, I wonder:
Q2: Does the Auslander-Buchsbaum formula holds for this depth (for an approriate definition of the dimension of $R$) ?
According to the cited papers, there seems to be no regular sequences defined for non-comm. rings. However, as far as I can see, the usual definition that $x_1,...,x_n \in R$ are $M$-regular, if $x_i$ is regular on $M/(\sum_{j=1}^iRx_i)M$ makes sense in the non-comm. case, too.
Q3: Why doesn't this definition work well resp. dosn't have nice properties ? (I believe it doesn't have nice properties because it isn't considered in the literature).
PS: There are more results for the depth of non-comm. graded rings in the literature, but I want to restrict to the ungraded case here.
[BT] J. Bueso, B. Torrecillas: Noncommutative local cohomology. Comm. Alg. 11(1983), 681-693
[R] J. Raynaud: Profondeur, hauteur et localisations en algebre non commutative. J. of Pure Appl. Alg. 31(1984), 199-215
