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If a and b are two nonzero elements of a ring R such that ab=0 then a and b are divisors of 0 or (0 divisors). In particular a is a left divisor of 0 and b is a right divisor of 0.

My question is if you have a0=0 then a is not a 0 divisor?

So it seems that 0 divisors will always come in pairs? So whenever you have a left 0 divisor, you automatically have a right 0 divisor.