In a general ring $A$ (commutative with $1$), should the sum of two zero divisors also a zero divisor? Could anyone give a proof or a countexample?
Moreover, consider the polynomial ring $A[x]$, where $A$ is a general ring, then could anyone ask the same question as above?
In fact, my question is: given two polynomials, both can be annihilated by some nonzero element in $A$, then is there exist some nonzero element in $A$ (or some nonzero element in $A[x]$ will also OK, in fact, the two cases are equivalent, see Introduction to Commutative Algebra, Atiyah and Macdonald, chapter 1, exercise 2) annihilate both the two polynomials? The essential is to annihilate the two ideals generated by these two elements simultaneously by an nonzero element.