# Question # 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.

Commutative Algebra
ANSWERED Let R and S be commutative rings. Prove that (a, b) is a zero-divisor in $$\displaystyle{R}\oplus{S}$$ if and only if a or b is a zero-divisor or exactly one of a or b is 0. 2020-12-15

Let (a,b) is a zero-divisor in $$\displaystyle{R}\oplus{S}$$
Since (a,b) is zero-divisor, there exists (c,d) in $$\displaystyle{R}\oplus{S}$$ such that
$$(a,b)(c,d)=(0,0)$$
$$\displaystyle\Rightarrow$$$$ac=0, bd=0$$
Since $$ac=0$$ $$\displaystyle\Rightarrow$$ a is a zero divisor in R or $$a=0$$
Also, $$bd = 0$$ $$\displaystyle\Rightarrow$$ 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 $$\displaystyle{c}\in{R}$$ such that ac=0
Also, b is zero divisor, there exist element $$\displaystyle{d}\in{S}$$ such that bd=0
$$\displaystyle\Rightarrow$$ $$ac\times bd=0$$
$$\displaystyle\Rightarrow$$ $$(a,b)\times (c,d)=(0,0)$$
Hence, for $$\displaystyle{\left({a},{b}\right)}\in{R}\oplus{S}$$, there exists (c,d) such that $$(a,b)\times(c,d)=(0,0)$$
By definition (a,b) is zero diisor in $$\displaystyle{R}\oplus{S}$$
Hence, proved