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.

Cheyanne Leigh 2020-12-14 Answered
Let R and S be commutative rings. Prove that (a, b) is a zero-divisor in RS if and only if a or b is a zero-divisor or exactly one of a or b is 0.
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Expert Answer

Tuthornt
Answered 2020-12-15 Author has 107 answers

Let (a,b) is a zero-divisor in RS
Since (a,b) is zero-divisor, there exists (c,d) in RS such that
(a,b)(c,d)=(0,0)
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 cR such that ac=0
Also, b is zero divisor, there exist element dS such that bd=0
ac×bd=0
(a,b)×(c,d)=(0,0)
Hence, for (a,b)RS, there exists (c,d) such that (a,b)×(c,d)=(0,0)
By definition (a,b) is zero diisor in RS
Hence, proved

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