26 Feb A closer look into a well-known quotient ring
The ring is quite a well- known ring in Algebra. Further, being a principal ideal domain, every ideal of is generated by a single element say and so can be taken in the form
We now consider the quotient ring and try to look for a precise way of representing the elements in it. First we suppose that and are both non zero. The quantities and deserve special attention in the discussion.
Result A: – An integer if and only if
Proof: – Since , we can write , where
Now let This implies that for some
, which gives and (“|” means divides.)
And since , so and which means that and are integers.
Also, say, where
Thus we get
which shows that
Conversely if then by using the relation it can be easily shown that
Therefore if and only if
Result B: – For any , if and only if
Proof: – Similar to above.
1) Since do not belong to so it is clear that the elements of are distinct.
2) Since so it is to be noted that
3) If then by division algorithm, there exist unique with such that Since so
Thus any element of the form , where , can be reduced to the form for a unique integer satisfying
The following result is further more generalized:
Proof: – The proof is a just a simple application of a well- known property of GCD.
We take any element of By division algorithm, we can write where Again since so there exist such that And it is easy to see that which gives
, where , (division algorithm)
where , (since )
Hence the result
Result D: – The elements in the set in RHS of D are distinct.
Proof: – Let
Let and be any two members of .
Since and a simple observation reveals that
1) cannot be a non –zero multiple of and
2) cannot be a non-zero multiple of
The above members will be equal if
i.e. is a multiple of d, but by (1) above i.e.
We thus have which in turn implies that is a multiple of but by (2) above, the only possibility is i.e. Thus the elements of are distinct.
Conclusion:- From the above results it is clear that
2) contains exactly the above elements.
3) If , then
which is isomorphic to the ring of integers modulo
4) Further if is prime then is a field and so is maximal.
5) Another way of representing is
6) It can be proved that the same result holds for the case when one of or is zero. In this case we can take and
There may be other better ways of representing the elements of
We can try looking for other such smarter ways !!
– Debashish Sharma, Junior Research Fellow, Dept. of Mathematics,NIT Silchar.