Ring homomorphism

In ring theory , special images between rings are considered , which are called ring homomorphisms. A ring homomorphism is a structure-preserving mapping between rings, and thus a special homomorphism .

definition

Given are two rings and . A function is called ring homomorphism if for all elements of : ${\displaystyle (R,+,\cdot )}$ ${\displaystyle (S,\oplus ,\otimes )}$ ${\displaystyle \varphi \colon \,R\to S}$ ${\displaystyle a,b}$ ${\displaystyle R}$

${\displaystyle \varphi (a+b)=\varphi (a)\oplus \varphi (b)}$ and ${\displaystyle \varphi (a\cdot b)=\varphi (a)\otimes \varphi (b).}$

The equation says that the homomorphism preserves the structure : It does not matter whether you first connect two elements and map the result, or first map the two elements and then connect the images.

Explanation

In other words, a ring homomorphism is a mapping between two rings that is both group homomorphism with respect to the additive groups of the two rings and semigroup homomorphism with respect to the multiplicative semigroups of the two rings.

For a "homomorphism of rings with one" there is usually an additional requirement. For example, the zero mapping from to is a ring homomorphism, but not a homomorphism of rings with one , since the special structure of the one is lost through the mapping: the one (like all other elements) becomes zero. ${\displaystyle \varphi (1_{R})=1_{S}}$ ${\displaystyle \mathbb {Z} }$ ${\displaystyle \mathbb {Z} }$

For a ring homomorphism the two sets are ${\displaystyle \operatorname {\varphi } }$

${\displaystyle \operatorname {Kern} \ \operatorname {\varphi } =\lbrace x\in R\mid \operatorname {\varphi } (x)=0\rbrace }$ and

${\displaystyle \operatorname {Bild} \ \operatorname {\varphi } =\operatorname {\varphi } (R)=\lbrace \operatorname {\varphi } (x)\in S\mid x\in R\rbrace }$

Are defined; from English and Latin one also writes ker instead of kernel and img instead of image , im or simply I (capital i). is a subring of , is an ideal in . A ring homomorphism is exactly then injective (ie a ring mono morphism) if applies. ${\displaystyle \operatorname {Bild} \ \operatorname {\varphi } }$ ${\displaystyle S}$ ${\displaystyle \operatorname {Kern} \ \operatorname {\varphi } }$ ${\displaystyle R}$ ${\displaystyle \operatorname {Kern} \ \operatorname {\varphi } =\lbrace 0\rbrace }$

Examples

The following figures are ring homomorphisms:

• The zero mapping ${\displaystyle f_{0}\colon R\to S;r\mapsto 0}$
• The inclusion picture for solid ${\displaystyle i\colon P\left(N\right)\to P\left(M\right);A\mapsto A}$ ${\displaystyle N\subsetneq M}$
• The complex conjugation ${\displaystyle \mathbb {C} \to \mathbb {C} ;z\mapsto {\bar {z}}}$
• The conjugation for a fixed entity ${\displaystyle f_{a}\colon R\to R;r\mapsto a\cdot r\cdot a^{-1}}$ ${\displaystyle a\in R^{*}}$
• ${\displaystyle \varphi \colon \,\mathbb {Z} \to \mathbb {Z} /{\mathit {n\mathbb {Z} }}}$or These are the remainder classes modulo n , the links of which are compatible with those from . ${\displaystyle {\mathit {z}}\mapsto {\mathit {z}}\operatorname {mod} {\mathit {n}}}$
  ${\displaystyle \mathbb {Z} }$

Individual evidence

1. ↑ Gerd Fischer : Linear Algebra . 14th revised edition. Vieweg, Wiesbaden 2003, ISBN 3-528-03217-0 , ( Vieweg Studium. Grundkurs Mathematik ), p. 145

literature

• Gerd Fischer: Linear Algebra. 14th revised edition. Vieweg, Wiesbaden 2003, ISBN 3-528-03217-0 , ( Vieweg study. Basic course in mathematics ).