Definition. algebra homomorphism [ca-001A]

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AlgHom

L∃∀N

Let \(A\) and \(B\) be \(R\)-algebras. \(\mathit {1}_A\) and \(\mathit {1}_B\) are ring homomorphisms from \(R\) to \(A\) and \(B\), respectively. A algebra homomorphism from \(A\) to \(B\) is a map \(f : \alpha \to _{a} \beta \) such that

1. \(f\) is a ring homomorphism
2. \(f(\mathit {1}_{A}(r)) = \mathit {1}_{B}(r)\) for each \(r \in R\)