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
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.
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)!