alternative proof of condition on a near ring to be a ring

Theorem 1.

Let $(R,+,\cdot)$ be a near ring with a multiplicative identity $1$ such that the $\cdot$ also left distributes over $+$ ; that is, $c\cdot(a+b)=c\cdot a+c\cdot b$ . Then $R$ is a ring.

Proof.

All that needs to be verified is commutativity of $+$ .

Let $a,b\in R$ . Consider the expression $(1+1)(a+b)$ .

We have:

$\displaystyle(1+1)(a+b)$	$\displaystyle=(1+1)a+(1+1)b$	$\displaystyle\quad\qquad\text{by left distributivity}$
	$\displaystyle=1a+1a+1b+1b$	$\displaystyle\quad\qquad\text{by right distributivity}$
	$\displaystyle=a+a+b+b$	$\displaystyle\quad\qquad\text{since }1\text{ is a multiplicative identity}$

On the other hand, we have:

	$\displaystyle(1+1)(a+b)$	$\displaystyle=1(a+b)+1(a+b)$	$\displaystyle\quad\qquad\text{by right distributivity}$
		$\displaystyle=a+b+a+b$	$\displaystyle\quad\qquad\text{since }1\text{ is a multiplicative identity}$

Thus, $a+a+b+b=a+b+a+b$ . Hence:

$\displaystyle a+b$	$\displaystyle=0+(a+b)+0$	$\displaystyle\quad\qquad\text{since }0\text{ is an additive identity ({http://% planetmath.org/AdditiveIdentity})}$
	$\displaystyle=(-a+a)+(a+b)+(b+-b)$	$\displaystyle\quad\qquad\text{by definition of additive inverse ({http://% planetmath.org/AdditiveInverse})}$
	$\displaystyle=-a+(a+a+b+b)+-b$	$\displaystyle\quad\qquad\text{by associativity of }+$
	$\displaystyle=-a+(a+b+a+b)+-b$	$\displaystyle\quad\qquad\text{since }a+a+b+b=a+b+a+b$
	$\displaystyle=(-a+a)+(b+a)+(b+-b)$	$\displaystyle\quad\qquad\text{by associativity of }+$
	$\displaystyle=0+(b+a)+0$	$\displaystyle\quad\qquad\text{by definition of }$
	$\displaystyle=b+a$	$\displaystyle\quad\qquad\text{since }0\text{ is an }$

∎

Title	alternative proof of condition on a near ring to be a ring
Canonical name	AlternativeProofOfConditionOnANearRingToBeARing
Date of creation	2013-03-22 17:20:06
Last modified on	2013-03-22 17:20:06
Owner	Wkbj79 (1863)
Last modified by	Wkbj79 (1863)
Numerical id	9
Author	Wkbj79 (1863)
Entry type	Proof
Classification	msc 20-00
Classification	msc 16-00
Classification	msc 13-00