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.

Question
Commutative Algebra
asked 2020-12-14
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.

Answers (1)

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.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
0

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