Exercise · ZW-ZERO-DIVISOR-010 · ★★★ Advanced
If \(a\neq0\) satisfies \(a^2=0\) in a ring, what is \(a\)?
Choose one option.
Show answer
- B. A zero divisor
Explanation
Since \(a\cdot a=0\) with \(a\neq0\), \(a\) is its own partner: a zero divisor. It is not a unit, since units annihilate nothing, and an idempotent satisfies \(a^2=a\), not \(a^2=0\) (except \(a=0\)).
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-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?