00:01
In the given question, we have to show that any ideal i, sorry, of the group z is of the form m such as that m times l, such as that l will belongs to the z.
00:25
Now to prove this, we will consider let i be any ideal of z.
00:34
Now if we have i is equal to 0 then we know that i is equal to 0.
00:44
Now suppose that i is not equal to 0 and let m be the smallest positive integer in z.
01:08
Then we have i is equal to m times l such that l will belongs to the z now since we have m belongs to the z and i is an ideal, then m times l will belong to the ideal for every integer.
01:38
L belongs to the z.
01:42
So we get m times l such as l, l belongs to the z is a subset of ideal i.
01:51
Now let any integer b belongs to the r.
01:58
Now if we have the value of b is equal to 0, then we get b is equal to m times 0, which belongs to the set m times l such as that l belongs to the set z.
02:13
Now if b is not equal to 0, then by euclidean algorithm, there is an integer.
02:37
And are such that, sorry, q and y, such that we have b is equal to m times q plus l, where 0 is less than or equals to l is less than or equals to m...