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Let me start recalling the Schanuel's Lemma:

If $M$ is a module and $P,P'$ are projective modules, then for every short exact sequences $0\to K\to P\to M\to 0$ and $0\to K'\to P'\to M\to 0$, there is an isomorphism $K\oplus P\cong P'\cong K'\oplus P'$P$. So, if you have a short exact sequence$0\to K\to P\to M\to 0$with$M$f.g.,$P$f.g. projective and$K$not f.g., you will not be able to find any other f.g. projective$P'$which admits a projection onto$M$with f.g. kernel. So, your question is equivalent to ask the following: for which class of rings does the class of f.g. left modules coincide with that of finitely presented left modules? The answer to this question is: the class of left Noetherian rings. In some cases, one can find such resolutions outside from the Noetherian context. For example, if a ring is left coherent you can find such resolutions for any finitely generated left ideal of the ring. 4 added 5 characters in body; deleted 12 characters in body Let me start recalling the Schanuel's Lemma: If$M$is a module and$P,P'$are projective modules, then for every short exact sequences$0\to K\to P\to M\to 0$and$0\to K'\to P'\to M$M\to 0$, there is an isomorphism $K\oplus P\cong K'\oplus P'$.

So, if you have a short exact sequence $0\to K\to P\to M\to 0$ with $M$ f.g., $P$ f.g. projective and $K$ not f.g., you will not be able to find any other finitely generated f.g. projective $P'$ which admits a projection to onto $M$ with f.g. kernel.

So, your question is equivalent to ask the following: for which class of rings does the class of f.g. left modules coincide with that of finitely presented left modules? The answer to this question is: the class of left Noetherian rings.

In some cases, one can find such resolutions outside from the Noetherian context. For example, if a ring is left coherent you can find such resolutions for any finitely generated left ideal of the ring.

3 deleted 3 characters in body

Let me start recalling the Schanuel's Lemma:

If $M$ is a module and $P,P'$ are projective modules, then for every short exact sequences $0\to K\to P\to M\to 0$ and $0\to K'\to P'\to M$, there is an isomorphism $K\oplus P\cong K'\oplus P'$.

So, if you have a short exact sequence $0\to K\to P\to M\to 0$ with $M$ and f.g., $P$ finitely generated f.g. projective and $K$ not f.g., you will not be able to find any other finitely generated projective $P'$ which admits a projection to $M$ with f.g. kernel.

So, your question is equivalent to ask the following: for which class of rings does the class of finitely generated f.g. left modules coincides coincide with that of finitely presented left modules? The answer of to this question is: the class of left Noetherian rings.

In some cases, one can find such resolutions outside from the Noetherian context. For example, if a ring is left coherent you can find such resolutions for any finitely generated left ideal of the ring.

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