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

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

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.

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

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