The following proofs of elementary ring properties use only the axioms that define a mathematical ring:
Basics
Multiplication by zero
Theorem: \( {\displaystyle 0a=a0=0} \)
\( {\displaystyle 0a=(0+0)a=(0a)+(0a)} \)
By subtracting (i.e. adding the additive inverse of) \( {\displaystyle 0a} \) on both sides of the equation, we get the desired result. The proof that \( {\displaystyle a0=0} \) is similar.
Zero ring
Theorem: A ring \( (R, +, \cdot) \)is the zero ring (that is, consists of precisely one element) if and only if 0 = 1 {\displaystyle 0=1} {\displaystyle 0=1}.
Suppose that \( {\displaystyle 1=0} \). Let a be any element in R; then \( {\displaystyle a=a\cdot 1=a\cdot 0=0} \). Therefore, \( (R, +, \cdot) \) is the zero ring. Conversely, if \( (R, +, \cdot) \) is the zero ring, it must contain precisely one element. Therefore, \( {\displaystyle 0 } \) and 1 is the same element, i.e. \( {\displaystyle 0=1}. \)
Multiplication by negative one
Theorem: \( {\displaystyle (-1)a=-a}
\( {\displaystyle (-1)\cdot a+a=(-1)\cdot a+1\cdot a=((-1)+1)\cdot a=0\cdot a=0} \)
Therefore \( {\displaystyle (-1)\cdot a=(-1)\cdot a+0=(-1)\cdot a+(a+(-a))=((-1)\cdot a+a)+(-a)=0+(-a)=(-a)}. \)
Multiplication by additive inverse
Theorem: \( {\displaystyle (-a)\cdot b=a\cdot (-b)=-(ab)} \)
To prove that the first expression equals the second one, \( {\displaystyle (-a)\cdot b=((-1)\cdot a)\cdot b=(a\cdot (-1))\cdot b=a\cdot ((-1)\cdot b)=a(-b).} \)
To prove that the first expression equals the third one, \( {\displaystyle (-a)\cdot b=((-1)\cdot a)\cdot b=(-1)\cdot (a\cdot b).} \)
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 \( {\displaystyle (-a)\cdot b} \) is indeed the inverse of \( {\displaystyle (a\cdot b)} \) by showing that adding them up results in the additive identity element,
\( {\displaystyle (a\cdot b)+(-a)\cdot b=(a-a)\cdot b=0\cdot b=0}. \)
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
Hellenica World - Scientific Library
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