The following proofs of elementary ring properties use only the axioms that define a mathematical ring:
Basics
Multiplication by zero
Theorem:
(Click "show" at right to see the proof of this theorem or "hide" to hide it.)
By subtracting (i.e. adding the additive inverse of)
![{\displaystyle 0\cdot a}](https://wikimedia.org/api/rest_v1/media/math/render/svg/116ce53fbe125c0f2bf7ee4190c34c5191103c5b)
on both sides of the equation, we get the desired result. The proof that
![{\displaystyle a\cdot 0=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/31f39efe37450f6bce024486f458a90d11a08278)
is similar.
Unique identity element per binary operation
Theorem: The identity element e for a binary operation (addition or multiplication) of a ring is unique.
(Click "show" at right to see the proof of this theorem or "hide" to hide it.)
If there is another identity element
![{\displaystyle e'}](https://wikimedia.org/api/rest_v1/media/math/render/svg/06c198f6710d781baaf94653df305a4881380033)
for the binary operation, then
![{\displaystyle e'a=ae'=a}](https://wikimedia.org/api/rest_v1/media/math/render/svg/84c97493f7940109cf8c43a6e6d952bcae58719b)
, and when
![{\displaystyle a=e}](https://wikimedia.org/api/rest_v1/media/math/render/svg/350efd647d49a06f61797e8613f2152545879282)
,
![{\displaystyle e'e=ee'=e=e'}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3017d4e7dab4ecf49f6e4159dbb2e2faa7a057f7)
where
![{\displaystyle ab}](https://wikimedia.org/api/rest_v1/media/math/render/svg/49337c5cf256196e2292f7047cb5da68c24ca95d)
is the binary operation on ring elements
![{\displaystyle a}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc)
and
![{\displaystyle b}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3)
.
Unique additive inverse element
Theorem: - a as the additive inverse element for a is unique.
(Click "show" at right to see the proof of this theorem or "hide" to hide it.)
If there is another inverse element
![{\displaystyle -a'}](https://wikimedia.org/api/rest_v1/media/math/render/svg/76387fc500469cbae63c118296c0f390a5d7a3e5)
for
![{\displaystyle a}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc)
, then
![{\displaystyle -a=-a+0=-a+a-a'=0-a'=-a'}](https://wikimedia.org/api/rest_v1/media/math/render/svg/03022c1c03b119d2f79e4069cf53770db3e0fcf0)
.
Unique multiplicative inverse element
Theorem: a−1 as the multiplicative inverse element for a is unique.
(Click "show" at right to see the proof of this theorem or "hide" to hide it.)
If there is another inverse element
![{\displaystyle a^{-1'}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/185f1e639f685540ff3944e06ebd248991d1ab9b)
for
![{\displaystyle a}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc)
, then
![{\displaystyle a^{-1}=a^{-1}\times 1=a^{-1}\times a\times a^{-1'}=1\times a^{-1'}=a^{-1'}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/39b71ad95d29eafa3ea514b72c4c30f1b9006699)
.
Zero ring
Theorem: A ring
is the zero ring (that is, consists of precisely one element) if and only if
.
(Click "show" at right to see the proof of this theorem or "hide" to hide it.)
Suppose that
![{\displaystyle 1=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/41647e2919d0c6740f9d118064d2dd244651ed60)
. Let
![{\displaystyle a}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc)
be any element in
![{\displaystyle R}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33)
; then
![{\displaystyle a=a\cdot 1=a\cdot 0=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/48debeacacde8982bbc54eefccaa6d77c492ee75)
. Therefore,
![{\displaystyle (R,+,\cdot )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e7a0f4d832c9b7871f68bc77313edbd25f82717e)
is the zero ring. Conversely, if
![{\displaystyle (R,+,\cdot )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e7a0f4d832c9b7871f68bc77313edbd25f82717e)
is the zero ring, it must contain precisely one element by its definition. Therefore,
![{\displaystyle 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950)
and
![{\displaystyle 1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf)
is the same element, i.e.
![{\displaystyle 0=1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a2bb1a7bcdbd7e274b7b8581e4357f09dbeee7fa)
.
Multiplication by negative one
Theorem:
(Click "show" at right to see the proof of this theorem or "hide" to hide it.)
Therefore
![{\displaystyle (-1)\cdot a=(-1)\cdot a+0=(-1)\cdot a+(a+(-a))=((-1)\cdot a+a)+(-a)=0+(-a)=(-a)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d415aa43b9aa221bfda887b38f46af513f320b0f)
.
Multiplication by additive inverse
Theorem:
(Click "show" at right to see the proof of this theorem or "hide" to hide it.)
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,
![{\displaystyle (a\cdot b)+(-a)\cdot b=(a-a)\cdot b=0\cdot b=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a283e5ca2938140be8beae64a2176f7860fe9519)
.