definition. ring homomorphism [chen2016infinitely, 4.5.1] [ca-0014]

✍️source

RingHomRingHomClassalgebraMap

L∃∀N

Let \((\alpha , +_\alpha , *_\alpha )\) and \((\beta , +_\beta , *_\beta )\) be rings. A ring homomorphism from \(\alpha \) to \(\beta \) is a map \(\mathit {1} : \alpha \to _{+*} \beta \) such that

1. \(\mathit {1}(x +_{\alpha } y) = \mathit {1}(x) +_{\beta } \mathit {1}(y)\) for each \(x,y \in \alpha \).
2. \(\mathit {1}(x *_{\alpha } y) = \mathit {1}(x) *_{\beta } \mathit {1}(y)\) for each \(x,y \in \alpha \).
3. \(\mathit {1}(1_{\alpha }) = 1_{\beta }\).