Additive mapping

Homomorphism $f:R_{1}\to R_{2}$ ${\ displaystyle f: R_ {1} \ to R_ {2}}$ ${\ displaystyle f: R_ {1} \ to R_ {2}}$ additive ring group $R_{1}$ ${\ displaystyle R_ {1}}$ $R_ {1}$ to additive ring group $R_{2}$ ${\ displaystyle R_ {2}}$ $R_ {2}$ called additive ring display $R_{1}$ ${\ displaystyle R_ {1}}$ $R_ {1}$ into the ring $R_{2}$ ${\ displaystyle R_ {2}}$ $R_ {2}$ .

According to the definition of an additive group homomorphism, the additive mapping $f$ ${\ displaystyle f}$ $f$ rings $R_{1}$ ${\ displaystyle R_ {1}}$ $R_ {1}$ into the ring $R_{2}$ ${\ displaystyle R_ {2}}$ $R_ {2}$ satisfies the property: $f(a+b)=f(a)+f(b)$ ${\ displaystyle f (a + b) = f (a) + f (b)}$ $f (a + b) = f (a) + f (b)$

It is not necessary that the additive mapping of the ring preserve the product.

If a $f$ ${\ displaystyle f}$ $f$ and $g$ ${\ displaystyle g}$ $g$ additive mapping, then mapping $f+g$ ${\ displaystyle f + g}$ $f + g$ additively. Similarly, adjectively mapping $a f b$ ${\ displaystyle afb}$ ${\ displaystyle afb}$ , if a $a,b\in R_{2}$ ${\ displaystyle a, b \ in R_ {2}}$ ${\ displaystyle a, b \ in R_ {2}}$ .

Additive body mapping

Let be $D$ ${\ displaystyle D}$ $D$ - body characteristics $0$ ${\ displaystyle 0}$ $0$ . Additive mapping

f:D\to D

{\ displaystyle f: D \ to D}

f:D\to D

bodies $D$ ${\ displaystyle D}$ $D$ can be represented as

f(x)={\sum _{s}}({}_{(s)0}f\ x\ {}_{(s)1}f)

{\ displaystyle f (x) = {\ sum _ {s}} ({} _ {(s) 0} f \ x \ {} _ {(s) 1} f)}

f(x)={\sum _{s}}({}_{(s)0}f\ x\ {}_{(s)1}f)

The number of items depends on the choice of function. $f$ ${\ displaystyle f}$ $f$ . Expressions ${}_{(s)0}f,{}_{(s)1}f\in D$ ${\ displaystyle {} _ {(s) 0} f, {} _ {(s) 1} f \ in D}$ ${}_{(s)0}f,{}_{(s)1}f\in D$ are called additive mapping components.

Additive mapping

Additive body mapping

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