Question:
Invertible elements in the ring of integers are
Invertible elements in the ring of integers are
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Invertible elements in a ring are called "Units." In $\mathbb{Z}$, the only units are $\pm 1$.
Updated On: Jul 3, 2026
- 1, -1
- 2, -2
- 3, -3
- 4, -4
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The Correct Option is A
Solution and Explanation
Step 1: Concept
An element 'a' in a ring is invertible (a unit) if there exists an element 'b' such that $a \cdot b = 1$.
Step 2: Meaning
In the ring of integers $\mathbb{Z}$, the multiplication is standard. We are looking for integers $x$ such that $1/x$ is also an integer.
Step 3: Analysis
For $1 \in \mathbb{Z}$, $1 \cdot 1 = 1$, so 1 is invertible. For $-1 \in \mathbb{Z}$, $(-1) \cdot (-1) = 1$, so -1 is invertible. For any other integer $|n| > 1$, $1/n$ is a fraction, not an integer.
Step 4: Conclusion
The only invertible elements in $\mathbb{Z}$ are 1 and -1. Final Answer: (A)
An element 'a' in a ring is invertible (a unit) if there exists an element 'b' such that $a \cdot b = 1$.
Step 2: Meaning
In the ring of integers $\mathbb{Z}$, the multiplication is standard. We are looking for integers $x$ such that $1/x$ is also an integer.
Step 3: Analysis
For $1 \in \mathbb{Z}$, $1 \cdot 1 = 1$, so 1 is invertible. For $-1 \in \mathbb{Z}$, $(-1) \cdot (-1) = 1$, so -1 is invertible. For any other integer $|n| > 1$, $1/n$ is a fraction, not an integer.
Step 4: Conclusion
The only invertible elements in $\mathbb{Z}$ are 1 and -1. Final Answer: (A)
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