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The following questions are from an assignment which I am trying to solve.

Prove that The following conditions on a ring R [with identity] are equivalent : (a) Every [unitary] R-module is projective. (b) Every short exact sequence of [unitary] R-modules is split exact. (c) Every [unitary] R-module is injective.

I have proved the rest but I am unable to prove (c) assuming (b) and will appreciate help.

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    $\begingroup$ What is your definition of injective module? $\endgroup$
    – azif00
    Aug 30, 2021 at 17:08
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    $\begingroup$ Well, whatever your definition is, an $R$-module $Q$ is injective if and only if every short exact sequence of the form $0 \longrightarrow Q \longrightarrow \bullet \longrightarrow \bullet \longrightarrow 0$ splits (prove it, if its necessary). Can you continue from here? $\endgroup$
    – azif00
    Aug 30, 2021 at 17:21

1 Answer 1

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Prove that The following conditions on a ring R [with identity] are equivalent :

(a) Every [unitary] R-module is projective.

(b) Every short exact sequence of [unitary] R-modules is split exact.

(c) Every [unitary] R-module is injective.

This is exercise 1 in Hungerford's Algebra (page 198). The solution is a direct consequence of Theorem 3.4 (page 192) and Proposition 3.13 (page 197). Let us see it in details.

Proof: (a)$\Rightarrow$(b). Suppose that every [unitary] R-module is projective. Given any short exact sequence $0 \to A \xrightarrow{f} B \xrightarrow{g} C \to 0$ of [unitary] R-modules, since $C$ is projective, we have, by Theorem 3.4, that
$0 \to A \xrightarrow{f} B \xrightarrow{g} C \to 0$ is split exact. So, every short exact sequence of [unitary] R-modules is split exact.

(b)$\Rightarrow$(a). Suppose that every short exact sequence of [unitary] R-modules is split exact. Given any R-module $C$, since every short exact sequence of [unitary] R-modules is split exact, we have that every short exact sequence $0 \to A \xrightarrow{f} B \xrightarrow{g} C \to 0$ is split exact. So, by Theorem 3.4, that $C$ is projective. So, every [unitary] R-module is projective.

The equivalence of (b) and (c) follows in a similar way, using Proposition 3.13, instead of Theorem 3.4.

(c)$\Rightarrow$(b). Suppose that every [unitary] R-module is injective. Given any short exact sequence $0 \to A \xrightarrow{f} B \xrightarrow{g} C \to 0$ of [unitary] R-modules, since $A$ is injective, we have, by Proposition 3.13, that
$0 \to A \xrightarrow{f} B \xrightarrow{g} C \to 0$ is split exact. So, every short exact sequence of [unitary] R-modules is split exact.

(b)$\Rightarrow$(c). Suppose that every short exact sequence of [unitary] R-modules is split exact. Given any R-module $A$, since every short exact sequence of [unitary] R-modules is split exact, we have that every short exact sequence $0 \to A \xrightarrow{f} B \xrightarrow{g} C \to 0$ is split exact. So, by Proposition 3.13, $A$ is injective. So, every [unitary] R-module is injective.

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  • $\begingroup$ Will check the answer in few days. Thanks! $\endgroup$
    – 3ibfwcbi
    Dec 16, 2021 at 12:28

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