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
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.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

Relevant Questions

Suppose that R and S are commutative rings with unites, Let PSJphiZSK be a ring homomorphism from R onto S and let A be an ideal of S.
a. If A is prime in S, show that $$\displaystyle\phi^{{-{{1}}}}{\left({A}\right)}={\left\lbrace{x}\in{R}{\mid}\phi{\left({x}\right)}\in{A}\right\rbrace}$$ is prime $$\displaystyle\in{R}$$.
b. If A is maximal in S, show that $$\displaystyle\phi^{{-{{1}}}}{\left({A}\right)}$$ is maximal $$\displaystyle\in{R}$$.
Suppose that a and b belong to a commutative ring and ab is a zero-divisor. Show that either a or b is a zero-divisor.
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Let A be nonepty set and P(A) be the power set of A. Recall the definition of power set:
$$\displaystyle{P}{\left({A}\right)}={\left\lbrace{x}{\mid}{x}\subseteq{A}\right\rbrace}$$
Show that symmetric deference operation on P(A) define by the formula
$$\displaystyle{x}\oplus{y}={\left({x}\cap{y}^{{c}}\right)}\cup{\left({y}\cap{x}^{{c}}\right)},{x}\in{P}{\left({A}\right)},{y}\in{p}{\left({A}\right)}$$
(where $$\displaystyle{y}^{{c}}$$ is the complement of y) the following statement istrue:
The algebraic operation o+ is commutative and associative on P(A).
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a) Prove that * is commutative, associate algebraic operation on $$\displaystyle\mathbb{Q}$$
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