# Thread: Units and Zero Divisors

1. ## Units and Zero Divisors

The question posed, is Prove a matrix A is a zero divisor if and only if it A is singular.
I have proved this statement.

The question is this true for all rings?
So i guess the question is for any ring R, An element a is a zero divisor if and only if it is not a unit.

===> Assume by contradiction a is a unit,
But a is a zero divisor, so there exist a b belong to R b not equal to 0 such that ab = 0
but we assume a is a unit so a^-1(ab) = (a^-1)0 = 0 which implies b = 0 which is a contradiction.

<== I having trouble with the reverse. We assume that A is not a unit. I am not sure where to go can someone help?

I almost positive that is it is true that if an element is unit then it is not a zero and likewise if an element is a 0 divisor then it is not a unit.

I am having problems showing if an element is a zero divisor then it cant be a unit

2. Originally Posted by john890
The question posed, is Prove a matrix A is a zero divisor if and only if it A is singular.
I have proved this statement.

The question is this true for all rings?
So i guess the question is for any ring R, An element a is a zero divisor if and only if it is not a unit.

===> Assume by contradiction a is a unit,
But a is a zero divisor, so there exist a b belong to R b not equal to 0 such that ab = 0
but we assume a is a unit so a^-1(ab) = (a^-1)0 = 0 which implies b = 0 which is a contradiction.

<== I having trouble with the reverse. We assume that A is not a unit. I am not sure where to go can someone help?

I almost positive that is it is true that if an element is unit then it is not a zero and likewise if an element is a 0 divisor then it is not a unit.

I am having problems showing if an element is a zero divisor then it cant be a unit

What you want to prove is false. Example, in $\mathbb{Z}$ , 2 is not a unit but it isn't azero divisor. Try the same with

any integral domain which isn't a field.

Tonio