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.bd=0

\(\displaystyle\Rightarrow\) (a,b).(c,d)=(0,0)

Hence, for \(\displaystyle{\left({a},{b}\right)}\in{R}\oplus{S}\), there exists (c,d) such that (a,b).(c,d)=(0,0)

By definition (a,b) is zero diisor in \(\displaystyle{R}\oplus{S}\)

Hence, proved

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.bd=0

\(\displaystyle\Rightarrow\) (a,b).(c,d)=(0,0)

Hence, for \(\displaystyle{\left({a},{b}\right)}\in{R}\oplus{S}\), there exists (c,d) such that (a,b).(c,d)=(0,0)

By definition (a,b) is zero diisor in \(\displaystyle{R}\oplus{S}\)

Hence, proved