Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In mathematics, especially in the area of abstract algebra known as module theory, a principal indecomposable module has many important relations to the study of a ring's modules, especially its simple modules, projective modules, and indecomposable modules. A (left) principal inde ...Full description
Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In mathematics, especially in the area of abstract algebra known as module theory, a principal indecomposable module has many important relations to the study of a ring's modules, especially its simple modules, projective modules, and indecomposable modules. A (left) principal indecomposable module of a ring R is a (left) submodule of R that is a direct summand of R and is an indecomposable module. Alternatively, it is an indecomposable, projective, cyclic module. Principal indecomposable modules are also called PIMs for short. The projective indecomposable modules over some rings have very close connections with those rings' simple, projective, and indecomposable modules. If the ring R is Artinian or even semiperfect, then R is a direct sum of principal indecomposable modules, and there is one isomorphism class of PIM per isomorphism class of simple module.