# Definition:R-Algebraic Structure Epimorphism

## 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 epimorphism if and only if:

$(1): \quad \phi$ is a surjection
$(2): \quad \forall k: k \in \left[{1 \,.\,.\, n}\right]: \forall x, y \in S: \phi \left({x \ast_k y}\right) = \phi \left({x}\right) \odot_k \phi \left({y}\right)$
$(3): \quad \forall x \in S: \forall \lambda \in R: \phi \left({\lambda \circ x}\right) = \lambda \otimes \phi \left({x}\right)$

This definition also applies to modules, and also to vector spaces.

## Linguistic Note

The word epimorphism comes from the Greek morphe (μορφή) meaning form or structure, with the prefix epi- meaning onto.

Thus epimorphism means onto (similar) structure.