Definition: Ring Homomorphism

Let \((R, +,\cdot)\) and \((S,\ast,\circ)\) be two rings. A function \(f:R\rightarrow S\) is called a ring homomorphism, if and only if for all \(x,y\in R\) the following equation holds:

\[\begin{array}{rcl} f(x + y)&=&f(x) \ast f(y),\\ f(x \cdot y)&=&f(x)\circ f(y) \end{array}\]

If the rings are unit rings, we require also

$$f(1_R)=1_S,$$ where $1_R$ and $1_S$ are the respective units of the both rings.

  1. Definition: Algebra over a Ring

Definitions: 1
Examples: 2
Lemmas: 3 4 5
Proofs: 6 7
Propositions: 8
Theorems: 9 10

Thank you to the contributors under CC BY-SA 4.0!




  1. Kramer Jürg, von Pippich, Anna-Maria: "Von den natürlichen Zahlen zu den Quaternionen", Springer-Spektrum, 2013

Adapted from CC BY-SA 3.0 Sources:

  1. Brenner, Prof. Dr. rer. nat., Holger: Various courses at the University of Osnabrück