On definitions and explicit examples of pure-injective modules

Question

I am interested in the following assumption on left $R$-modules: for a module $I$ and all injective homomorphisms $A\to B$ of finitely generated (or possibly finitely presented) modules I want the homomorphism $Hom_R(B,I)\to Hom_R(A,I)$ to be surjective. Is this condition strictly weaker than the injectivity of $I$; how can one construct examples of this sort?

What is the relation of my condition to pure injectivity of $R$-modules? I do not understand its relation to the "standard" definition of the latter notion; also, what is the relation of the "standard" definition to Terminology 11.1 in the paper "Relative Homological Algebra and Purity in Triangulated Categories" of Beligiannis?

Edit. Thanks to the answer by Leonid Positselski, now I know that the term I need is "fp-injectve", whereas "pure injective" probably means something else.

Answer 1

This is not pure-injectivity. The relevant concept is that of an fp-injective ("finitely presented-injective") module, otherwise known as an "absolutely pure" module. A left $R$-module $J$ is said to be fp-injective if for any finitely presented left $R$-module $M$ one has $Ext^1_R(M,J)=0$.

The notion of an fp-injective left $R$-module is particularly well-behaved when the ring $R$ is left coherent. Over a left Noetherian ring $R$, fp-injectivity is equivalent to injectivity.

The class of all fp-injective left $R$-modules is always closed under infinite direct sums (while the class of all injective left $R$-modules is closed under infinite direct sums if and only if the ring $R$ is left Noetherian). Thus infinite direct sums of injective left $R$-modules are typical examples of fp-injective left $R$-modules that are not injective. When $R$ is left coherent, the class of all fp-injective left $R$-modules is also closed under (filtered) direct limits.

References:

1. B.Stenström, "Coherent rings and $FP$-injective modules", Journ. London Math. Soc. vol.2, 1970, https://doi.org/10.1112/jlms/s2-2.2.323

2. C.Megibden, "Absolutely pure modules", Proc. Amer. Math. Soc. vol.26, 1970, https://doi.org/10.1090/S0002-9939-1970-0294409-8

3. my paper L.Positselski "Coherent rings, fp-injective modules, dualizing complexes, and covariant Serre-Grothendieck duality", Selecta Math. vol.23, 2017, https://arxiv.org/abs/1504.00700