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Question Number 190784 by mnjuly1970 last updated on 11/Apr/23

$$$$$$\mathrm{If},0⇢{\mathrm{M}}^{\prime }\stackrel{f}{⇢}\mathrm{M}\stackrel{g}{⇢}{\mathrm{M}}^{″}⇢0\mathrm{is}$$$$\mathrm{a}\mathrm{short}\mathrm{exact}\mathrm{sequence}\mathrm{and}{\mathrm{M}}^{\prime },{\mathrm{M}}^{″}\mathrm{are}$$$$\mathrm{two}\mathrm{finitely}\mathrm{generated}\mathrm{R}-\mathrm{modules}$$$$\mathrm{then}\mathrm{prove}\mathrm{M}\mathrm{is}\mathrm{finitely}\mathrm{generated}.$$$$\mathrm{Hint}:f,garetwoR-homomorphism.$$

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