Definition:R-Algebraic Structure Monomorphism

Definition
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 monomorphism :


 * $(1): \quad \phi$ is an injection
 * $(2): \quad \forall k: k \in \closedint 1 n: \forall x, y \in S: \map \phi {x \ast_k y} = \map \phi x \odot_k \map \phi y$
 * $(3): \quad \forall x \in S: \forall \lambda \in R: \map \phi {\lambda \circ x} = \lambda \otimes \map \phi x$.

That is, :


 * $(1): \quad \phi$ is an injection
 * $(2): \quad \phi$ is an $R$-algebraic structure homomorphism.

This definition continues to apply when $S$ and $T$ are modules, and also when they are vector spaces.

Also see

 * Definition:Monomorphism (Abstract Algebra)