Definition:R-Algebraic Structure Homomorphism

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 $\phi$ is an $R$-algebraic structure homomorphism iff:

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

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

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

Sources

• Seth Warner: Modern Algebra (1965): $\S 28$