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