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 if and only if:

$(1): \quad \phi$ 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, if and only if:

$(1): \quad \phi$ 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.

Vector Space Monomorphism

Let $V$ and $W$ be $K$-vector spaces.

Then $\phi: V \to W$ is a vector space monomorphism if and only if:

$(1): \quad \phi$ is an injection

$(2): \quad \forall \mathbf x, \mathbf y \in V: \map \phi {\mathbf x + \mathbf y} = \map \phi {\mathbf x} + \map \phi {\mathbf y}$

$(3): \quad \forall \mathbf x \in V: \forall \lambda \in K: \map \phi {\lambda \mathbf x} = \lambda \map \phi {\mathbf x}$

Also see

• Definition:Monomorphism (Abstract Algebra)

Linguistic Note

The word monomorphism comes from the Greek morphe (μορφή) meaning form or structure, with the prefix mono- meaning single.

Thus monomorphism means single (similar) structure.

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {V}$: Vector Spaces: $\S 28$. Linear Transformations