# Let R and S be commutative rings. Prove that (a, b) is a zero-divisor in R o+ S if and only if a or b is a zero-divisor or exactly one of a or b is 0.

Let R and S be commutative rings. Prove that (a, b) is a zero-divisor in $R\oplus S$ if and only if a or b is a zero-divisor or exactly one of a or b is 0.
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Tuthornt

Let (a,b) is a zero-divisor in $R\oplus S$
Since (a,b) is zero-divisor, there exists (c,d) in $R\oplus S$ such that
$\left(a,b\right)\left(c,d\right)=\left(0,0\right)$
$⇒$$ac=0,bd=0$
Since $ac=0$ $⇒$ a is a zero divisor in R or $a=0$
Also, $bd=0$ $⇒$ b is zero divisor in R or $b=0$
Conversely:
Let a and b are zero divisors.
Since a is zero divisor,there exist element $c\in R$ such that ac=0
Also, b is zero divisor, there exist element $d\in S$ such that bd=0
$⇒$ $ac×bd=0$
$⇒$ $\left(a,b\right)×\left(c,d\right)=\left(0,0\right)$
Hence, for $\left(a,b\right)\in R\oplus S$, there exists (c,d) such that $\left(a,b\right)×\left(c,d\right)=\left(0,0\right)$
By definition (a,b) is zero diisor in $R\oplus S$
Hence, proved