# Category:Ring Homomorphisms

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This category contains results about **Ring Homomorphisms**.

Definitions specific to this category can be found in Definitions/Ring Homomorphisms.

Let $\struct {R, +, \circ}$ and $\struct{S, \oplus, *}$ be rings.

Let $\phi: R \to S$ be a mapping such that both $+$ and $\circ$ have the morphism property under $\phi$.

That is, $\forall a, b \in R$:

\(\text {(1)}: \quad\) | \(\ds \map \phi {a + b}\) | \(=\) | \(\ds \map \phi a \oplus \map \phi b\) | |||||||||||

\(\text {(2)}: \quad\) | \(\ds \map \phi {a \circ b}\) | \(=\) | \(\ds \map \phi a * \map \phi b\) |

Then $\phi: \struct {R, +, \circ} \to \struct {S, \oplus, *}$ is a ring homomorphism.

## Subcategories

This category has the following 8 subcategories, out of 8 total.

### F

### R

- Ring Endomorphisms (1 P)
- Ring Epimorphisms (13 P)
- Ring Monomorphisms (8 P)

## Pages in category "Ring Homomorphisms"

The following 24 pages are in this category, out of 24 total.

### C

### F

### K

### P

### R

- Ring Homomorphism by Idempotent
- Ring Homomorphism from Ring with Unity to Integral Domain Preserves Unity
- Ring Homomorphism of Addition is Group Homomorphism
- Ring Homomorphism Preserves Negatives
- Ring Homomorphism Preserves Subrings
- Ring Homomorphism Preserves Subrings/Corollary
- Ring Homomorphism Preserves Zero