TITLE: Definition of an algebra homomorphism
QUESTION [1 upvotes]: Our definition of an algebra homomorphism $f:A_1\to A_2$ doesn't requires $f(1_{A_1})=1_{A_2}$. Is this automatic? I don't think so, so why one doesn't include this in the definition?

REPLY [0 votes]: Consider the $\mathbb{Z}$-algebras $\mathbb{R}$ and $A=\mathbb{R}\times\mathbb{R}$ (the product in $A$ is componentwise). Note that $1_A=(1,1)$ and $f:\mathbb{R}\to\mathbb{R}\times\mathbb{R}$ difined by $f(x)=(x,0)$ is homomorfism of algebras, but $(1,0)=f(1)\neq 1_A$.
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