الجبر Injective Module

[Princess Moon]

a direct product of ​

R​

-modules is injective iff each one is injective but I need an example to show that the direct sum of injective modules need not be injective.​

the ​

Bass-Papp Theorem​

asserts that a commutative ring ​

R​

is Noetherian iff every direct sum of injective ​

R​

-modules is injective. Thus every non-Noetherian ring carries a counterexample.​

​

if​

I1⊊I2⊊…⊊In⊊…​

is an infinite properly ascending chain of ideals of R, then for all n let En=E(R/In) be the injective envelope of R/In​

, and let ​

E=⨁∞n=1En​

. Then ​

E​

is a direct sum of injective modules and that ​

E​

is not itself injective.​

How to show that E is not injective???
^_~ ​