A triangulated category is said to be indecomposable if it admits no nontrivial semiorthogonal decompositions. We introduce a definition of noncommutatively stably semiorthogonally (NSSI) variety. This propery implies, among other things, that each smooth proper subvariety has derived coherent sheaves, and $Y$ NSSI, then for any variety $X$ all decompositions $X \times Y$ are induced from $X$. ...

We develop a criterion for coalition formation among goal-directed agents, the indecomposable do-ut-des property. The indecomposable do-ut-des property refines the do-ut-des property (literally give to get) by considering the fact that agents prefer to form coalitions whose components cannot be formed independently. A formal description of this property is provided as well as an analysis of alg...

Let R be local Noetherian ring of depth at least two. We prove that there are indecomposable R-modules which are free on the punctured spectrum of constant, arbitrarily large, rank.