Definition:R-Algebraic Structure Homomorphism

Definition

Let $R$ be a ring.

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.

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

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

$(1): \quad \forall k \in \closedint 1 n: \forall x, y \in S: \map \phi {x \ast_k y} = \map \phi x \odot_k \map \phi y$

$(2): \quad \forall x \in S: \forall \lambda \in R: \map \phi {\lambda \circ x} = \lambda \otimes \map \phi x$

where $\closedint 1 n = \set {1, 2, \ldots, n}$ denotes an integer interval.

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

Also see

• Definition:Homomorphism (Abstract Algebra)

• Results about $R$-algebraic structure homomorphisms can be found here.

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.

Sources

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