Exercise · ZW-ZERO-DIVISOR-006 · ★★☆ Intermediate
Which statement is true in a ring \(R\) with identity, \(R\neq0\)?
Choose one option.
Show answer
- B. A unit is never a zero divisor
Explanation
If \(u\) is a unit and \(uv=0\), then \(v=u^{-1}uv=0\): a unit is never a zero divisor. The others fail: \(\mathbb{Z}_7\) has no zero divisors, 0 is excluded by definition, and \(ab=0\) does not force \(a=0\) or \(b=0\) in general.
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-007 In ℤ×ℤ, which of these elements is a zero divisor?
- 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?