Page 157 - DMTH403_ABSTRACT_ALGEBRA

Abstract Algebra

Notes
Example: Let H = ( a + bi + cj + dk | a, b, c, d  R ), where i, j, k are symbols that satisfy
i = – 1 = j = k , ij = k = – ji, jk = i = – kj ki = j = – ik.
2
2
2
We define addition and multiplication in H by
(a + bi + cj + dk) + (a + b i + c j + d k )
l
i
i
1
= (a + a ) (b + b ) + (c – c ) t (d + d )k, and
1 i
1
1 j
1
(a + bi + cj + dk) (a + b i + cj+ d k) = (aa – bb – cc – dd ) + (ab + ha + cd – dc ) + (ac – bd + ca 1
1
1
1
1
1 i
1
l
1
l
1
l
1
1
+ db )j – (ad + bc – cb + da )k
1
1
1
1
1
(This multiplication may seem complicated. But it is not so. It is simply performed as for
polynomials, keeping the relationships between i, j and k in mind.)
Show that H is a ring.
Solution: Note that ( ± 1, ± i, ± j, ± k ] is the group QH.
Now, you can verify that (H, +) is an abelian group in which the additive identity is 0 = 0 + 0i +
0j + 0k, multiplication in H is associative, the distributive laws hold and
I = 1 + 0i + 0j + 0k is the unity in H.
Do you agree that H is not a commutative ring? You will if You remember that ij  ji, for
example.
So far, in this unit we have discussed various types of rings. We have seen examples of commutative
and non-commutative rings. Though non-commutative rings are very important for the sake of
simplicity we shall only deal with commutative rings henceforth. Thus, from now on, for us a
ring will always mean a commutative ring. We would like you to remember that both + and .
are commutative in a commutative ring.
Now, let us summarise what we have done in this unit.
Self Assessment
1. For each a in R. There exists X in R such that a + X = :0 = ................ i.e. every elements of R
has an additive inverse.
(a) a . x (b) x + a
(c) x + a (d) a + x
-1
-1
2. If a and b are integers, we say that a is congruent to b modulo n : f ................ is divisible
by n.
(a) a + b (b) a – b
(c) a . b (d) a/b
3. A × B is with unity if A and B are with unity. If A and B have identies e and e respectively,
2
1
then the identity of A × B is ................
(a) e + e 2 (b) e + e 1  1
1
2
1
1
(c) (e, e ) (d) (e ,e )


2
1
2
4. The ................ for addition and multiplication and the generalised distributive law.
(a) law of indices (b) Ring
(c) Subring (d) ideal
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