Remark

In Theorem 1, left ideals may be replaced by right ideals. In Chapter VIII, § 5, no. 2, we shall study non-zero rings $\mathbf{A}$ which have no two-sided ideal distinct from 0 and $\mathbf{A}$; such rings (called quasi-simple) are not necessarily fields (for example, the ring $\mathbf{M}_2(\mathbf{Q})$ of square matrices of order 2 with rational coefficients is quasi-simple but is not a field).