On ring homomorphisms (Q1293804)

Let \({\mathbb R}[x]\) be the algebra of all polynomials in one indeterminate with real coefficients and \((x)\) be the ideal in \({\mathbb R}[x]\) generated by \(x\). It is shown that for each subring \(B\) of \({\mathbb R}\) containing \({\mathbb Q}\) there is a ring homomorphism from \((x)\) onto \(B\). From this it follows that all subfields of \({\mathbb R}\) containing \({\mathbb Q}\) are residual fields of \((x)\), not every prime ideal of \((x)\) is maximal, not all homomorphisms \((x)\to {\mathbb R}\) are \({\mathbb R}\)-linear (even if they are surjective) and \((x)\) is not a Noetherian ring.