Proofs of elementary ring properties

The following proofs of elementary ring properties use only the axioms that define a mathematical ring:

BasicsEdit

Multiplication by zeroEdit

Theorem:  

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By subtracting (i.e. adding the additive inverse of)   on both sides of the equation, we get the desired result. The proof that   is similar.

Zero ringEdit

Theorem: A ring   is the zero ring (that is, consists of precisely one element) if and only if  .

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Suppose that  . Let   be any element in  ; then  . Therefore,   is the zero ring. Conversely, if   is the zero ring, it must contain precisely one element. Therefore,   and   is the same element, i.e.  .

Multiplication by negative oneEdit

Theorem:  

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Therefore  .

Multiplication by additive inverseEdit

Theorem:  

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To prove that the first expression equals the second one,  

To prove that the first expression equals the third one,  

A pseudo-ring does not necessarily have a multiplicative identity element. To prove that the first expression equals the third one without assuming the existence of a multiplicative identity, we show that   is indeed the inverse of   by showing that adding them up results in the additive identity element,

 .