# Definition:R-Algebraic Structure Monomorphism

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## Definition

Let $\left({S, \ast_1, \ast_2, \ldots, \ast_n, \circ}\right)_R$ and $\left({T, \odot_1, \odot_2, \ldots, \odot_n, \otimes}\right)_R$ 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
$(2): \quad \forall k: k \in \left[{1 \,.\,.\, n}\right]: \forall x, y \in S: \phi \left({x \ast_k y}\right) = \phi \left({x}\right) \odot_k \phi \left({y}\right)$
$(3): \quad \forall x \in S: \forall \lambda \in R: \phi \left({\lambda \circ x}\right) = \lambda \otimes \phi \left({x}\right)$.

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
$(2): \quad \forall \mathbf x, \mathbf y \in V: \phi \left({\mathbf x + \mathbf y}\right) = \phi \left({\mathbf x}\right) + \phi \left({\mathbf y}\right)$
$(3): \quad \forall \mathbf x \in V: \forall \lambda \in K: \phi \left({\lambda \mathbf x}\right) = \lambda \phi \left({\mathbf x}\right)$

## 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.