A is an algebra over a field F.

Let 'a' in A have the following property:

a.b = 0 for all 'b' in A.

Does this imply a=0? If yes why?

Thanks,

Aman

PS: I was trying the above for a ring. I am kind of convinced that in rings 'a' need not be equal to 0. Haven't been able to find a concrete example though.