Either prove or give a counterexample: If R is a ring with zero divisors, then R[x] is a ring with zero divisors? (where R[x] is the polynomial ring with coefficients from R and R can be any ring) And...

Either prove or give a counterexample:

If R is a ring with zero divisors, then R[x] is a ring with zero divisors? (where R[x] is the polynomial ring with coefficients from R and R can be any ring)

And also vice-versa.. If R a ring and R[x] contains zero divisors, then R contains zero divisors