CONSTRUCTION OF MODULES WITH A PRESCRIBED DIRECT SUM DECOMPOSITION

We give some criteria for recognizing local rings that allow us to show that indecomposable AB5* modules over commutative rings and couniform modules over noetherian commutative rings have a local endomorphism ring. We also develop some theory on methods to construct modules with a prescribed direct-sum decomposition. As an application we realize an interesting class of commutative monoids as monoids of direct summands of a direct sum of a countable number of copies of a suitable artinian cyclic module, showing that there may appear a rich supply of direct summands that are not a direct sum of artinian modules. An important gadget for proving our realization result is a variation of a method for realizing a given ring as the endomorphism ring of a cyclic (artinian) module due to Armendariz, Fisher and Snider.