Morphism of Rings

definitionring-theory

Definition

For rings $(R,+,\cdot )$ and $(S,+,\cdot )$, $f:(R,+,\cdot )\to (S,+,\cdot )$ is a morphism if,

1. $f(r+r^{\prime })=f(r)+f(r^{\prime })$ for all $r,r^{\prime }\in R$.
2. $f(r\cdot r^{\prime })=f(r)\cdot f(r^{\prime })$ for all $r,r^{\prime }\in R$.

Additionally,

• If $R,S$ are unital ring, we say $f$ is a morphism of unital rings if additionally $f(1_R)=1_S$.
• If $f$ is a bijective morphism, say $f$ is an isomorphism.
• If there exists an isomorphism $f:R\to S$, say $R\simeq S$.
• If $f:R\to R$ is an isomorphism, say $f$ is an automorphism.

Properties

• $f(-x)=-f(x)$
• $f(n_{\mathbb{Z}}\cdot x)=n_{\mathbb{Z}}\cdot f(x)$ for all $n\in \mathbb{Z}$.
• $f(x^n)=(f(x){)}^n$ for all $n\ge 1$.
• If $x$ is invertible for multiplication (in a unital ring), then there exists $y=x^{-1}$, such that $yx=xy=1_R$, and $f$ is a morphism of unital rings, then $f(x)$ invertible, $(f(x){)}^{-1}=f(x^{-1})$.

Examples