Ring Homomorphism Preserves Subrings/Corollary
Jump to navigation
Jump to search
Corollary to Ring Homomorphism Preserves Subrings
Let $\struct {R_1, +_1, \circ_1}$ and $\struct {R_2, +_2, \circ_2}$ be rings.
The image of a ring homomorphism $\phi: R_1 \to R_2$ is a subring of $R_2$.
Proof
From Null Ring and Ring Itself Subrings, $R_1$ is a subring of itself.
The result then follows from Ring Homomorphism Preserves Subrings.
$\blacksquare$
Sources
- 1970: B. Hartley and T.O. Hawkes: Rings, Modules and Linear Algebra ... (previous) ... (next): $\S 2.2$: Homomorphisms: Lemma $2.6 \ \text{(ii)}$
- 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 57.3$ Ring homomorphisms: $\text{(i)}$