we show that every semi-artinian module which is contained in a direct sum of finitely presented modules in $si[m]$, is weakly co-semisimple if and only if it is regular in $si[m]$. as a consequence, we observe that every semi-artinian ring is regular in the sense of von neumann if and only if its simple modules are $fp$-injective.

We prove that if a quadratic system satisses the direct sum conjecture strongly in the quadratic algorithm model, then it satisses the direct sum conjecture strongly in the straight line algorithm model. Therefore, if the strong direct sum conjecture is true for the quadratic algorithm model then it is also true for the straight line algorithm model. We use this to classify all the minimal prog...

A Direct Sum Theorem holds in a model of computation, when for every problem solving some k input instances together is k times as expensive as solving one. We show that Direct Sum Theorems hold in the models of deterministic and randomized decision trees for all relations. We also note that a near optimal Direct Sum Theorem holds for quantum decision trees for boolean functions.