On the existence of left and right almost split morphisms

We discuss the existence of left and right almost split morphisms for a skeletally small category ℳ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429187759/559656af-5888-4439-a5d2-08d7a38d7161/content/inq_chapter1_1_1.tif"/> of modules over an arbitrary ring R. To this end, we associate to ℳ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429187759/559656af-5888-4439-a5d2-08d7a38d7161/content/inq_chapter1_1_2.tif"/> a certain R-module M and investigate finiteness conditions on M viewed as a module over its endomorphism ring.