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In this question let us assume that $R$ is a ring without unity.

Here are two conditions for an $R$-module $J$.

(a). $J$ is injective, that is, given any $R$-module monomorphism $g:A\to B$ and a homomorphism $f:A\to J$, there is an $R$-module homomorphism $h:B\to J$ such that $f=h\circ g$.

(b). Every exact sequence $ 0 \to J \to A \to B \to 0 $ of $R$-modules splits.

Question. Does (b) imply (a)?

If $R$ had a unity then the answer is yes; what if $R$ has no unity? I'm working with the exercise IV.3.1 in Algebra by Hungerford. It asks to prove that

Every short exact sequence of $R$-modules splits if and only if every $R$-module is injective.

Then it might be helpful if the answer to my question is yes.

In this book, a ring may not have a unity. The definition of injective module also does not require for a ring to have unity.

Springfield
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    What is your definition of "injective"? There are many equivalent definitions. – Arturo Magidin May 18 '21 at 17:31
  • A book talking about injectives/projectives in rings without identity would not be impossible, but highly improbable. Please double check your context to see if it isn't an unnecessarily exotic case to consider. – rschwieb May 18 '21 at 20:51

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