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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 $K\oplus P\cong K'\oplus P'$.

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 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.

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 $K\oplus P'\cong K'\oplus P$.

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 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.

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 $M$ f.g., $P$ 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 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.

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 $K\oplus P\cong K'\oplus P'$.

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 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.

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 $P$ finitely generated 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 the class of finitely generated left modules coincides with finitely presented left modules? The answer of 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.

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 $M$ f.g., $P$ 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 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.