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Unit 14: Rings
14.2 Elementary Properties Notes
In this section we will prove some simple but important properties of rings which are immediate
consequences of the definition of a ring. As we go along you must not forget that for any ring R,
(R, +) is an abelian group. Hence, the results obtained for groups in the earlier units are applicable
to the abelian group (R, +). In particular,
(i) the zero element, 0, and the additive inverse of any element is unique.
(ii) the cancellation law holds for addition; i.e., a, b, c R , a + c = b + c a = b.
As we have mentioned earlier, we will write a b for a + (b) and ab for a. b, where a, b R.
So let us state some properties which follow from the axiom R6, mainly.
Theorem 1: Let R be a ring. Then, for any a, b, c R,
(i) a0 = 0 = 0a,
(ii) a(b) = (a)b = (ab),
(iii) ( a) ( b) = ab,
(iv) a(b c) = ab ac, and
Proof:
(i) Now, 0 + 0 = 0
a(0 + 0) = a0
a0 + a0 = a0, applying the distributive law.
= a0 + 0, since 0 is the additive identity.
a0 = 0, by the cancellation law for (R, +).
Using the other distributive law, we can similarly show that 0a = 0.
Thus, a0 = 0 = 0a for all a R.
(ii) From the definition of additive inverse, we know that b + ( b) = 0.
Now, 0 = a0, from (i) above.
= a(b + ( b)), as 0 = b + ( b).
= ab + a( b), by distributivity.
Now, ab + [ (ab)] = 0 and ab + a( b) = 0. But you know that the additive inverse of an
element is unique.
Hence, we get (ab) = a( b).
In the same manner, using the fact that a + (a) = 0, we get (ab) = ( a)b.
Thus, a( b) = ( a)b = (ab) for all a, b R.
(iii) For a, b R,
( a) ( b) = (a( b)), from (ii) above.
= a( ( b)), from (ii) above.
= ab, since b is the additive inverse of ( b).
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