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We give an algorithm for working out the indecomposable direct summands in aKrull--Schmidt decomposition of a tensor product of two simple modules forG=SL_3 in characteristics 2 and 3. It is shown that there is a finite family ofmodules such that every such indecomposable summand is expressible as a twistedtensor product of members of that family.  Along the way we obtain the submodule structure of various Weyl and tiltingmodules. Some of the tilting modules that turn up in characteristic 3 are notrigid; these seem to provide the first example of non-rigid tilting modules foralgebraic groups. These non-rigid tilting modules lead to examples of non-rigidprojective indecomposable modules for Schur algebras, as shown in the Appendix.  Higher characteristics (for SL_3) will be considered in a later paper.

Decomposition of tensor products of modular irreducible representations for $SL_3$ (With an Appendix by C.M. Ringel)