ABSTRACT
In this chapter we shall introduce the notion of pure exact sequences with respect to a class P https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203755532/9b5423eb-c97a-47c6-b834-ac464e69e878/content/eq1292.tif"/> of modules, generalizing splitting sequences. We already know that injective, semisimple and projective modules may be characterized by the splitting of certain exact sequences. Similarly we shall consider modules distinguished by the P https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203755532/9b5423eb-c97a-47c6-b834-ac464e69e878/content/eq1293.tif"/> -purity of certain short exact sequences (see § 35). Choosing in R-MOD as P https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203755532/9b5423eb-c97a-47c6-b834-ac464e69e878/content/eq1294.tif"/> the class of all finitely presented ii-modules we obtain the usual notion of purity in R-MOD.
