00:04
Hello, in this question we have to answer some questions about rings.
00:10
The first question is what is characteristic of a ring? now given any ring r, there is always a map, there is a unique map phi r, unique homomorphism phi r from the ring of integers to r taking 1 to the multiplicative identity of r or 1 of r.
00:40
Then the kernel of phi r is either 0 or it is equal to a principal ideal generated by a positive number for some positive number n.
00:59
Then the characteristic of r is then defined to be 0 if this phi r is injective or it is equal to n if the kernel of phi r is equal to this principal ideal for a positive integer n.
01:23
Or in more, this is the formal definition, but more easily if n times the multiplicative identity of r is equal to 0, then the smallest positive integer such that this thing holds that is n times identity is equal to 0 and is called the characteristic of a ring.
01:49
The smallest positive integer such that n times this thing is equal to 0 is called the characteristic of a ring.
01:57
And if 1 times r is never equal to 0 for all positive integers, in that case we call the characteristic to be 0.
02:07
So this is the definition of characteristic of a ring.
02:13
Now the second question is when is r called an integral domain? now r is an integral domain if and only if r has the following property.
02:30
A, b belong to r and a, b equal to 0.
02:35
This implies a equal to 0 or b equal to 0.
02:40
If a ring r has this property, then the ring r is called an integral domain.
02:48
In part c, we have to prove that the characteristic of integral domain is prime or 0.
02:55
Ok, suppose r is an integral domain is an integral domain and the characteristic of r is a times b where a and b are positive integers and a is not equal to 1 and b is not equal to 1.
03:21
Then we have that a times 1 times b times 1 is equal to ab times 1 is equal to 0.
03:36
But this means that this element times this element, the product of these two elements is 0.
03:43
But it's an integral domain.
03:45
This implies that either a times 1 is equal to 0 or b times 1 is equal to 0.
03:51
In both, since a and b are non -zero, sorry, a is not equal to 0 here and b is not equal to 0.
04:00
No, a is not equal to 1.
04:06
I'm sorry.
04:10
A is not equal to 1 and b is not equal to 1.
04:16
So that ab is a proper, this is a proper factorization of the product ab.
04:23
Ok, then we must have that a times 1 is equal to 0 or b times 1 is equal to 0.
04:29
But in note that because it's a proper factorization, this implies that a is less than ab and b is also less than ab.
04:40
A and b, both of them, each of them are less than ab.
04:44
But if a dot 1 is equal to 0 and b dot 1 is equal to 0, then it implies that there is a number less than ab such that that number times 1 equal to 0.
04:57
But this violates that ab is the characteristic of the ring.
05:04
But this implies that ab is not the characteristic of ring.
05:11
It violates this property...