1) Let \( R \) and \( S \) denote nonzero rings. Prove that the ring \( R \times S\) is not a field.
Proof. Choose \( 0 \neq r \in R, 0 \neq s \in S \). Then \( (r,0) \cdot (0,s) = (0,0) \) in \( R \times S \), but
\( (r,0) \neq 0 \) in \( R \times S \) and \( (0,s) \neq 0 \) in \( R \times S \). That is, \( R \times S \) has zero-divisors,
but no field does. Therefore \( R \times S \) is not a field. QED.
Note: This is Ex. §7.6,#4 in Dummit & Foote.