Remark

Most of the properties proved in this no. for subspaces of a vector space do not hold for submodules of a free module whose dimension ($§ 2,$ no. 7, Remark 2) is defined. For example, an ideal of a commutative ring does not necessarily admit a basis, for there are integral domains $A$ in which certain ideals are not principal (VII, $§ 1,$ no. 1) and any two elements of such a ring are linearly dependent ($§ 1,$ no. 11, Remark 1). A submodule of a free $A$-module $E$ may be free, distinct from $E$ and have the same dimension as $E$, as is shown by principal ideals in an integral domain $A$; the same example proves moreover that a free submodule of a free $A$-module does not necessarily admit a supplement.