It's sort of like the inverse function theorem, and that is why it is so strong. If you have n functions vanishing at the origin of k^n and want to know if they give a local coordinate system, you ask if their differentials are independent at the origin. Or equivalently if their differentials generate the cotangent space at the origin. So in a [not necessarily noetherian, thanks Georges!] local ring (O,m), Nakayama's lemma says you can detect that elements of the maximal ideal generate that ideal, hence act sort of like coordinate functions, just by knowing their differentials, i.e. their residues in the Zariski cotangent space m/m^2, generate that linear space.

Those versions of the lemma you linked to are almost unrecognizable forms of this simple statement, but that's the way abstract math goes as we know. But the idea is the same, you have a hypotheses about a truncated version of your statement, and you get out the fuller version. The Jacobson radical stuff is there to disguise the fact that it doesn't say much unless you are in a local setting. I.e. in a local ring the Jacobson radical is pretty big and you get a better result. In a polynomial ring with tiny Jacobson radical you get nothing.

It's sort of like the inverse function theorem, and that is why it is so strong. If you have $n$ functions vanishing at the origin of $k^n$ and want to know if they give a local coordinate system, you ask if their differentials are independent at the origin. Or equivalently if their differentials generate the cotangent space at the origin. So in a [not necessarily noetherian, thanks Georges!] local ring $(\mathcal{O},\mathfrak{m})$, Nakayama's lemma says you can detect that elements of the maximal ideal generate that ideal, hence act sort of like coordinate functions, just by knowing their differentials, i.e. their residues in the Zariski cotangent space $\mathfrak{m}/\mathfrak{m}^2$, generate that linear space.

Those versions of the lemma you linked to are almost unrecognizable forms of this simple statement, but that's the way abstract math goes as we know. But the idea is the same, you have a hypotheses about a truncated version of your statement, and you get out the fuller version. The Jacobson radical stuff is there to disguise the fact that it doesn't say much unless you are in a local setting. I.e. in a local ring the Jacobson radical is pretty big and you get a better result. In a polynomial ring with tiny Jacobson radical you get nothing.

It's sort of like the inverse function theorem, and that is why it is so strong. If you have n functions vanishing at the origin of k^n and want to know if they give a local coordinate system, you ask if their differentials are independent at the origin. Or equivalently if their differentials generate the cotangent space at the origin. So in a noetherian local ring (O,m), Nakayama's lemma says you can detect that elements of the maximal ideal generate that ideal, hence act sort of like coordinate functions, just by knowing their differentials, i.e. their residues in the Zariski cotangent space m/m^2, generate that linear space.

Those versions of the lemma you linked to are almost unrecognizable forms of this simple statement, but that's the way abstract math goes as we know. But the idea is the same, you have a hypotheses about a truncated version of your statement, and you get out the fuller version. The Jacobson radical stuff is there to disguise the fact that it doesn't say much unless you are in a local setting. I.e. in a local ring the Jacobson radical is pretty big and you get a better result. In a polynomial ring with tiny Jacobson radical you get nothing.

It's sort of like the inverse function theorem, and that is why it is so strong. If you have n functions vanishing at the origin of k^n and want to know if they give a local coordinate system, you ask if their differentials are independent at the origin. Or equivalently if their differentials generate the cotangent space at the origin. So in a [not necessarily noetherian, thanks Georges!] local ring (O,m), Nakayama's lemma says you can detect that elements of the maximal ideal generate that ideal, hence act sort of like coordinate functions, just by knowing their differentials, i.e. their residues in the Zariski cotangent space m/m^2, generate that linear space.

Those versions of the lemma you linked to are almost unrecognizable forms of this simple statement, but that's the way abstract math goes as we know. But the idea is the same, you have a hypotheses about a truncated version of your statement, and you get out the fuller version. The Jacobson radical stuff is there to disguise the fact that it doesn't say much unless you are in a local setting. I.e. in a local ring the Jacobson radical is pretty big and you get a better result. In a polynomial ring with tiny Jacobson radical you get nothing.