〈 Berger | Dillon 〉

R-modules which are subsets of R are extremely interesting; they were among the first modules to be studied and continue to play a central role in modern algebra. In contrast, K-vector spaces that are subsets of K are not so interesting. There are only two, zero and K itself.

R-modules that are subsets of R are called ideals; a field has no nontrivial proper ideals. Ideals are interesting as the kernels of ring homomorphisms; since fields have only two this means that field morphisms (which preserve the unit) are all invective.