Constant Mapping to Identity is Homomorphism/Rings
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Theorem
Let $\struct {R_1, +_1, \circ_1}$ and $\struct {R_2, +_2, \circ_2}$ be rings with zeroes $0_1$ and $0_2$ respectively.
Let $\zeta$ be the zero homomorphism from $R_1$ to $R_2$, that is:
- $\forall x \in R_1: \map \zeta x = 0_2$
Then $\zeta$ is a ring homomorphism whose image is $\set {0_2}$ and whose kernel is $R_1$.
Proof
The additive groups of $\struct {R_1, +_1, \circ_1}$ and $\struct {R_2, +_2, \circ_2}$ are $\struct {R_1, +_1}$ and $\struct {R_2, +_2}$ respectively.
Their identities are $0_1$ and $0_2$ respectively.
Thus from the Constant Mapping to Group Identity is Homomorphism we have that $\zeta: \struct {R_1, +_1} \to \struct {R_2, +_2}$ is a group homomorphism:
- $\map \zeta {x +_1 y} = \map \zeta x +_2 \map \zeta y$
Then we have:
\(\ds \map \zeta {x \circ_1 y}\) | \(=\) | \(\ds 0_2\) | as $x \circ_1 y \in R_1$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \map \zeta x \circ_2 \map \zeta y\) | as $\map \zeta x = 0_2$ and $\map \zeta y = 0_2$ |
The results about image and kernel follow directly by definition.
$\blacksquare$
Sources
- 1964: Iain T. Adamson: Introduction to Field Theory ... (previous) ... (next): Chapter $\text {I}$: Elementary Definitions: $\S 3$. Homomorphisms: Example $1$
- 1969: C.R.J. Clapham: Introduction to Abstract Algebra ... (previous) ... (next): Chapter $5$: Rings: $\S 24$. Homomorphisms: Example $42$