One of the main problems in the representation theory of an artin algebra Λ is determining when two modules M and N in mod Λ are isomorphic. Λ completely general answer to this problem was given in Chapter VI in terms of R-lengths of the R-module of morphisms from each indecomposable module to the given module M. At first sight it seems hopeless to ever use this criterion for proving that two modules are isomorphic. Our use of it in this chapter shows that this is not entirely true. Nonetheless, it is obvious that it is desirable to have other criteria which are more manageable, even if they are not as general.

In this chapter we concentrate on giving conditions on a pair of indecomposable Λ-modules which guarantee that they are isomorphic in terms of such familiar invariants as their composition factors, their projective presentations or their tops and socles. The basic assumption is that one or both of the modules do not lie on certain types of cycles of morphisms in mod Λ, namely short cycles. We start the chapter by discussing these types of modules.

Short cycles

In this section we introduce and study the notions of short cycles and short chains.