The depth of V as an R-module is defined in that paper to be the least positive integer n such that V?" ?" ?" ?" ?" V ( n times V, tensor product of R-modules, diagonal action ) has the same constituent indecomposable modules as V ?" ?" ?" ?" ?" V ( n + 1 times V ) ( not counting multiplicities ).
22.
In the finite-length situation, decomposition into indecomposables is particularly useful, because of the Krull-Schmidt theorem : every finite-length module can be written as a direct sum of finitely many indecomposable modules, and this decomposition is essentially unique ( meaning that if you have a different decomposition into indecomposable, then the summands of the first decomposition can be paired off with the summands of the second decomposition so that the members of each pair are isomorphic ).
23.
While R is not isomorphic to M, R ?" R is isomorphic to M ?" M; thus the images of the M summands give indecomposable submodules L 1, L 2 < R ?" R which give a different decomposition of R ?" R . The failure of uniquely factorizing R ?" R into a direct sum of indecomposable modules is directly related ( via the ideal class group ) to the failure of the unique factorization of elements of R into irreducible elements of R.
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