Definition:R-Algebraic Structure Homomorphism

Definition

Let $R$ be a ring.

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.

Let $\phi: S \to T$ be a mapping.

Then $\phi$ is an $R$-algebraic structure homomorphism if and only if:

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

where $\left[{1 \,.\,.\, n}\right] = \left\{{1, 2, \ldots, n}\right\}$ denotes an integer interval.

Note that this definition also applies to modules and vector spaces.

Linguistic Note

The word homomorphism comes from the Greek morphe (μορφή) meaning form or structure, with the prefix homo- meaning similar.

Thus homomorphism means similar structure.