Thread: 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

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.

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.

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.

Are you 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.

Originally Posted by Drexel28

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

Are you 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.

first of all not all rings without zero divisor is an integral domain! think of a ring that is not commutative.
then I am not asking to prove an statement by another statement, for example: a room has window, a room with window is light.
now asking how do we know a room has window, to show it is light. in response it is not convenient to say if a window is light it has window because how do we know it is light? then comes up then we one can say we know it because it has window!

same question: a ring without zero divisor has property that its zero ideal is prime ideal, and this property holds if a ring doesn't have zero divisor. you can also do the same thing with other properties about ring with or without zero divisors. and this is all I have been facing through algebra.

now how can I be sure that there is no zero divisor in the ring? (suppose a ring that is not easy for us to determine all its element)!