Let \(k\) be a field and let \(L/k\) and \(L'/k\) be field extensions. A \(k\)-embedding \(L\embeds L'\) is a ring homomorphism \(\tau\colon L\to L'\) that restricts to the identity on \(k\):
Here \(\iota\) and \(\iota'\) are the inclusions, and the triangle commutes: \(\tau\circ\iota=\iota'\).
More generally, let \(\sigma\colon k\to k'\) be a homomorphism of fields and let \(L/k\) and \(L'/k'\) be field extensions with inclusions \(\iota\colon k\to L\) and \(\iota'\colon k'\to L'\). A lift of \(\sigma\) to \(L\to L'\) is a ring homomorphism \(\tau\colon L\to L'\) such that \(\tau\circ\iota=\iota'\circ\sigma\):
A lift of \(\sigma=\id_k\) is a \(k\)-embedding.
Every ring homomorphism \(\tau\colon L\to L'\) between fields is injective: \(\ker\tau\) is an ideal of \(L\) not containing \(1\), and the only such ideal of a field is \(0\).