/sci/ - Science & Math

Who the fuck was Nagata and why is every single counterexample in algebra attributed to him

Let [math]R[/math] be a commutative ring with identity, and [math]\phi:R^m\to R^n[/math] a module homomorphism. If [math]\mathfrak m[/math] is a maximal ideal, then we have an induced vector space homomorphism [math]\phi_{\mathfrak m}: R^m/\mathfrak m R^m\to R^n/\mathfrak m R^n[/math].

In particular, this can be used to show that if [math]\phi[/math] is an isomorphism, then [math]m=n[/math], and if [math]\phi[/math] a surjection, then [math]m\geq n[/math], by basic dimension theory. However, the same argument cannot be used to show [math]m\leq n[/math] if [math]\phi[/math] is an injection. Why?