Exercise · ZW-ZERO-DIVISOR-007 · ★★☆ Intermediate
In \(\mathbb{Z}\times\mathbb{Z}\), which of these elements is a zero divisor?
Choose one option.
Show answer
- D. \((2,0)\)
Explanation
Since \((2,0)(0,1)=(0,0)\) with both factors nonzero, \((2,0)\) is a zero divisor. \((1,1)\) is a unit, \((3,5)(a,b)=0\) only when \(a=b=0\), and \((0,0)\) is excluded by definition.
Review the concept: Zero divisor
A zero divisor of a ring \(R\) is an element \(a\neq0\) for which there is another element \(b\neq0\) of \(R\) whose product with \(a\) is zero.
\[ab=0,\quad a\neq0,\ b\neq0\]
More practice problems: zero divisor
- ZW-ZERO-DIVISOR-001 In ℤ_12, which of these elements is a zero divisor?
- ZW-ZERO-DIVISOR-002 How many zero divisors does ℤ_10 have?
- ZW-ZERO-DIVISOR-003 Which of these rings has no zero divisors?
- ZW-ZERO-DIVISOR-004 In ℤ_15, which b≠0 satisfies 6b=0?
- ZW-ZERO-DIVISOR-005 In the ring of 2×2 real matrices, which of these matrices is a zero divisor?
- ZW-ZERO-DIVISOR-006 Which statement is true in a ring R with identity, R≠0?
- ZW-ZERO-DIVISOR-008 In ℤ_8, which pair with x≠ y satisfies 4x=4y?
- ZW-ZERO-DIVISOR-009 How many zero divisors does ℤ_21 have?