Composite of Homomorphisms is Homomorphism

Theorem

Algebraic Structures

Let:

$\struct {S_1, \otimes_1, \otimes_2, \ldots, \otimes_n}$
$\struct {S_2, *_1, *_2, \ldots, *_n}$
$\struct {S_3, \oplus_1, \oplus_2, \ldots, \oplus_n}$

Let:

$\phi: \struct {S_1, \otimes_1, \otimes_2, \ldots, \otimes_n} \to \struct {S_2, *_1, *_2, \ldots, *_n}$
$\psi: \struct {S_2, *_1, *_2, \ldots, *_n} \to \struct {S_3, \oplus_1, \oplus_2, \ldots, \oplus_n}$

Then the composite of $\phi$ and $\psi$ is also a homomorphism.

R-Algebraic Structures

Let $\struct {R, +_R, \times_R}$ be a ring.

Let:

$\struct {S_1, \odot_1, \ldots, \odot_n}$
$\struct {S_2, {\odot'}_1, \ldots, {\odot'}_n}$
$\struct {S_3, {\odot}_1, \ldots, {\odot}_n}$

be algebraic structures each with $n$ operations.

Let:

$\struct {S_1, *_1}_R$
$\struct {S_2, *_2}_R$
$\struct {S_3, *_3}_R$

Let:

$\phi: \struct {S_1, *_1}_R \to \struct {S_2, *_2}_R$
$\psi: \struct {S_2, *_2}_R \to \struct {S_3, *_3}_R$

Then the composite of $\phi$ and $\psi$ is also a homomorphism.