\begin{definition}[Definition:R-Algebraic Structure Epimorphism]
Let $\struct {S, \ast_1, \ast_2, \ldots, \ast_n, \circ}_R$ and $\struct {T, \odot_1, \odot_2, \ldots, \odot_n, \otimes}_R$ be $R$-algebraic structures.
Then $\phi: S \to T$ is an '''$R$-algebraic structure epimorphism''' {{iff}}:
:$(1): \quad \phi$ is a surjection
:$(2): \quad \forall k: k \in \closedint 1 n: \forall x, y \in S: \map \phi {x \ast_k y} = \map \phi x \odot_k \map \phi y$
:$(3): \quad \forall x \in S: \forall \lambda \in R: \map \phi {\lambda \circ x} = \lambda \otimes \map \phi x$
This definition also applies to modules, and also to vector spaces.
\end{definition}