Elementary Properties of Ring

Some basic elementary properties of a ring can be illustrated with help of following theorems and these properties are used in developing further concepts in rings and these properties are building of rings.

Theorem:
If R is a ring, then for all a,bare in R.
(a) a \cdot 0 = 0 \cdot a = a
(b) a\left( { - b} \right) = \left( { - a} \right)b  = - \left( {ab} \right)
(c) \left( { - a} \right)\left( { - b} \right) = ab

Proof: (a) We know that
a0 = a\left( {0 + 0} \right) = a0 + a0\,\,\,\forall a \in  R\,\,\,\,\,\,\,\left[ {{\text{using}}\,{\text{distributive law}}} \right]
Since R is a group under addition, applying right cancellation law,
a0 = a0 + a0 \Rightarrow a + a0 = a0 + a0 \Rightarrow a0 =  0
 Similarly, 0a = \left( {0 + 0} \right)a = 0a + 0a\,\,\,\forall  a \in R\,\,\,\,\,\,\,\left[ {{\text{using}}\,{\text{distributive law}}} \right]
\therefore \,\,\,0 + 0a = 0a + 0a\,\,\,\,\,\,\,\left[  {{\text{because}}\,0 = 0a + 0a} \right]
Applying right cancellation law for addition, we get 0 = 0a i.e. 0a = 0
Thus a0 = 0a = 0

Proof: (b) To prove that a\left( { - b} \right) = - ab we should show that ab = a\left( { - b} \right) = 0
We know that a\left[  {b + \left( b \right)} \right] = a0 = 0 because b + \left( { - b} \right) = 0 with the above result (a)
ab + a\left( { - b} \right) = 0\,\,\,\,\,\,\,\left[  {{\text{by}}\,{\text{distributive}}\,{\text{law}}} \right]
\therefore \,\,\,a\left( { - b} \right) = - \left( {ab} \right)
Similarly, to show \left(  { - a} \right)b = - ab, we must show that ab + \left( { - a} \right)b = 0
But ab + \left( { -  a} \right)b = \left[ {a + \left( { - a} \right)} \right]b = 0b = 0
\therefore \,\,\, - \left( a \right)b = - \left( {ab} \right) hence the result

Proof: (c) Actually to prove \left( { - a} \right)\left(  { - b} \right) = ab is a special case of forgoing article. However its proof is given as under:
\left( { - a} \right)\left( { - b} \right) = - \left[ {a\left( { - b} \right)} \right] =  \left[ { - \left( {ab} \right)} \right] = ab
Because  - \left( { -  x} \right) = x is a consequence of the fact that in a group inverse of the inverse of an element is element itself. 

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