Chapter 5.1, Problem 46E

Let $R$

be a set of elements containing the unity $e$, that satisfy all of the conditions in

Definition $5.1$

a, except condition $5$

: Addition is commutative. Prove that condition $5$ must

also hold.

Definition $5.1$

a $\square$

Definition of a Ring

Suppose $R$

is a set in which a relation of equality, denoted by $=$, and operations of addition and multiplication, denoted by $+$

and $\cdot$, respectively, are defined. Then $R$

is a ring (with respect to these operations) if the following conditions are satisfied:

$R$

is closed under addition: $x\in R$ and $y\in R$

imply $x+y\in R$.

Addition in $R$

is associative: $x+(y+z)=(x+y)+z$ for all $x,y,z$ in $R$.

$R$ contains an additive identity $0$

: $x+0=0+x=x$ for all $x\in R$.

$R$ contains an additive inverse: For $x$ in $R$, there exists $-x$ in $R$ such that

$x+(-x)=(-x)+x=0$.

Addition in $R$ is commutative: $x+y=y+x$ for all $x,y$ in $R$.

$R$

is closed under multiplication: $x\in R$ and $y\in R$ imply $x\cdot y\in R$.

Multiplication in $R$

is associative: $x\cdot (y\cdot z)=(x\cdot y)\cdot z$ for all $x,y,z$

in $R$.

Two distributive laws hold in $R$

: $x\cdot (y+z)=x\cdot y+x\cdot z$ and

$(x+y)\cdot z=x\cdot z+y\cdot z$ for all $x,y,z$

in $R$.

The notation $xy$

will be used interchageably with $x\cdot y$ to indicate

multiplication.