Homomorphs., Monomorphs., and Isomorphs. Between Algebras

# Homomorphisms, Monomorphisms, and Isomorphisms Between Algebras

Definition: Let $\mathfrak{A}$ and $\mathfrak{B}$ be algebras over $\mathbb{F}$ and let $\phi \in \mathcal L(\mathfrak{A}, \mathfrak{B})$. Then:1) $\phi$ is a Homomorphism from $\mathfrak{A}$ to $\mathfrak{B}$ if $\phi(a_1a_2) = \phi(a_1)\phi(a_2)$ for all $a_1, a_2 \in \mathfrak{A}$.2) $\phi$ is a Monomorphism from $\mathfrak{A}$ to $\mathfrak{B}$ if it is a homomorphism that is injective.3) $\phi$ is an Isomorphism from $\mathfrak{A}$ to $\mathfrak{B}$ if it is a homomorphism that is bijective, and if an isomorphism from $\mathfrak{A}$ to $\mathfrak{B}$ exists then $\mathfrak{A}$ and $\mathfrak{B}$ are said to be Isomorphic. |