\begin{definition}[Definition:R-Algebraic Structure Homomorphism]
Let $R$ be a ring.
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.
Let $\phi: S \to T$ be a mapping.
Then $\phi$ is an '''$R$-algebraic structure homomorphism''' {{iff}}:
:$(1): \quad \forall k \in \closedint 1 n: \forall x, y \in S: \map \phi {x \ast_k y} = \map \phi x \odot_k \map \phi y$
:$(2): \quad \forall x \in S: \forall \lambda \in R: \map \phi {\lambda \circ x} = \lambda \otimes \map \phi x$
where $\closedint 1 n = \set {1, 2, \ldots, n}$ denotes an integer interval.
Note that this definition also applies to modules and vector spaces.
\end{definition}