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I will summarize some of the main results in a recent paper of Ivo Herzog.

Every module may be purely embedded in a unique, minimal, pure-injective module called the pure-injective envelope. For instance, the abelian group of the integers localized at a prime p embeds purely in the group of p-adic numbers. This pure-injective envelope also has a ring structure, and as such it is the endomorphism ring of Z localized at p.

The general situation is much more complicated, but under certain (reasonable) circumstances, a double dual of a ring R gives the pure-injective envelope, which itself has a ring structure.

I will go over some of the highlights of Herzog's paper and try to give you the general flavour of a rather difficult series of results.

Applications of duality to the pure-injective envelope, I. Herzog, Algebras and Representation Theory, to appear.