Selforthogonal modules without obvious direct summands

We describe a sufficient condition which explains the aboundance of many rather small, and not necessarily faithful, selforthogonal modules M with the following properties: (1)The projective (resp. injective) dimension of M is finite and bigger than 1 . (2) The intersection of the kernel of all covariant (resp. contravariant Hom and Ext functors associated to M is equal to zero. As we shall see, both "discrete" properties (that is properties of indecomposable projective-injective modules), or "continuous" properties (that is properties of more or less "visible" classes of modules) play an important role. Mopreover, a kind of "cancellation" strategy of the obvious direct summands of both tilting and cotilting modules leads to rather small modules M , sdatisfying (1) and (2), and defined over finite dimensional algebras of both finite and infinite Representation Type. We also show that left and/or right "cancellations", as well as "additions" and "deformations" of bounded complexes (of projectives module) turn out to be useful building blocks in the construction of bounded complexes (of projective modules) with completely different properties