Question:
The characteristic of the ring \((\mathbb{Z}_6,+_6,\times_6)\) is
The characteristic of the ring \((\mathbb{Z}_6,+_6,\times_6)\) is
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The characteristic of \(\mathbb{Z}_n\) is \(n\).
Updated On: Jul 3, 2026
- \(3\)
- \(4\)
- \(5\)
- \(6\)
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The Correct Option is D
Solution and Explanation
Concept:
The characteristic of a ring is the least positive integer \(n\) such that \[ n\cdot 1=0 \] where \(1\) is the multiplicative identity of the ring.
Step 1: Identify the ring. The ring is \[ \mathbb{Z}_6 \] In \(\mathbb{Z}_6\), all calculations are done modulo \(6\). The identity element is \[ 1 \]
Step 2: Find the least positive \(n\). We need the least positive integer \(n\) such that \[ n\cdot 1\equiv 0\pmod 6 \] This means \[ n\equiv 0\pmod 6 \] The least positive such \(n\) is \[ 6 \]
Step 3: Final answer. \[ \boxed{6} \]
The characteristic of a ring is the least positive integer \(n\) such that \[ n\cdot 1=0 \] where \(1\) is the multiplicative identity of the ring.
Step 1: Identify the ring. The ring is \[ \mathbb{Z}_6 \] In \(\mathbb{Z}_6\), all calculations are done modulo \(6\). The identity element is \[ 1 \]
Step 2: Find the least positive \(n\). We need the least positive integer \(n\) such that \[ n\cdot 1\equiv 0\pmod 6 \] This means \[ n\equiv 0\pmod 6 \] The least positive such \(n\) is \[ 6 \]
Step 3: Final answer. \[ \boxed{6} \]
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