Proving bx=c Can Never Have a Single Solution in R

[silvermane]

1. The problem statement:

Let R be a commutative ring, and suppose b is a zero divisor in R. Let c e any element in R. Prove that the equation bx = c can never have a single solution in R.
(meaning that if it has a solution, it will have more than 1)

3. My attempt at a proof:
Well, I'm stuck as to the interpretation, and where to even begin for that matter. If you have any hints/tips that could get me started, that would be great. I don't want an answer, just a little scaffolding to get me there.

Any input is greatly appreciated. It means a lot when people help

 

[VeeEight]

b is a zero divisor seems to be important. Try using that by left multiplying bx by non zero y where yb = 0.

 

[silvermane]

b is a zero divisor seems to be important. Try using that by left multiplying bx by non zero y where yb = 0.

Yeah this is very helpful! Thank you!