# zero divisor

• Apr 8th 2010, 09:21 PM
hamidr
zero divisor
Hi, I am looking for a general way to be able to find if a ring has zero divisor or not.

I know what the definition is, but the definition doesnt provide a general method to define if there is zero divisor. for example, z(root(-5)), how do you check if there is a zero divisor
• Apr 8th 2010, 09:26 PM
Drexel28
Quote:

Originally Posted by hamidr
Hi, I am looking for a general way to be able to find if a ring has zero divisor or not.

I know what the definition is, but the definition doesnt provide a general method to define if there is zero divisor. for example, z(root(-5)), how do you check if there is a zero divisor

There are many different equivalent definitions of a ring being an integral domain. For example, that the zero ideal is prime.
• Apr 8th 2010, 09:44 PM
hamidr
Quote:

Originally Posted by Drexel28
There are many different equivalent definitions of a ring being an integral domain. For example, that the zero ideal is prime.

Drexel how do I check that? how do I check if zero*any element is in zero ideal, then there is no other element that * any element is in zero ideal

the problem that I am facing in abstract algebra , is that everything is being said equivalence to each other, but then there is no method given to check any of those equivalent relations. it is true that if zero ideal is not prime, then we will have only zero as the element in the ideal therefore zero is the only zero in the ring. but how can you be sure there it is prime ideal?
and I am assuming the ring that have zero divisor are not domains, so that they are not integral domain nor a PID or UFD.
• Apr 8th 2010, 09:52 PM
Drexel28
Quote:

Originally Posted by hamidr
Drexel how do I check that? how do I check if zero*any element is in zero ideal, then there is no other element that * any element is in zero ideal

the problem that I am facing in abstract algebra , is that everything is being said equivalence to each other, but then there is no method given to check any of those equivalent relations. it is true that if zero ideal is not prime, then we will have only zero as the element in the ideal therefore zero is the only zero in the ring. but how can you be sure there it is prime ideal?
and I am assuming the ring that have zero divisor are not domains, so that they are not integral domain nor a PID or UFD.

Ok, so I guess this becomes a question of what you're asking.

A) How do you prove that the satement I made is actually equivalent to being an integral domain

B) How do I in practice prove that something is an integral domain.
• Apr 8th 2010, 10:15 PM
hamidr
Quote:

Originally Posted by Drexel28
Ok, so I guess this becomes a question of what you're asking.