Just as in group theory, the lattice isomorphism theory holds for rings modulo ideals. Specifically, there is an inclusion preserving bijection between subrings of R that contain I and the set of subrings of R/I. Furthermore, A is an ideal containing I in R if and only if A/I is an ideal in R/I. In other words, the lattice composed of subrings that contain I and the lattice of R/I "look" exactly the same. The proof of this is rather tedious, but I'm told not difficult.

Now, armed with this fact and my theorem in post #3, the proof follows right away. M is maximal if and only if there aren't any nontrivial ideals in R containing M, excluding M itself. There aren't any nontrivial ideals in R containing M excluding M itself if and only if there aren't any nontrivial ideals in R/M. Finally, there aren't any nontrivial ideals in R/M if and only if R/M is a field.

]]>More specifically, when solving systems of n-variable polynomial equations over the complex field, there exists a solution if the polynomials form a proper ideal in C[x_1, ..., x_n].

]]>]]>A ring R is a field if and only if

itsonly ideals are 1 and R.