Exercise · ZW-FIRST-ISOMORPHISM-THEOREM-FOR-RINGS-003 · ★☆☆ Basic
How many elements does the kernel of the homomorphism \(\varphi\colon \mathbb{Z}_{12}\to \mathbb{Z}_{4}\), \(\varphi(a)=a \bmod 4\), have?
Choose one option.
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- A. \(3\)
Explanation
The kernel is \(\{0,4,8\}\), the elements of \(\mathbb{Z}_{12}\) divisible by 4. It matches \(12/4=3\), since \(\varphi\) is surjective.
Review the concept: First isomorphism theorem for rings
If \(\varphi\colon R\to S\) is a ring homomorphism with kernel \(\ker\varphi\), the quotient ring of \(R\) by \(\ker\varphi\) is isomorphic to the image of \(\varphi\).
\[R/\ker\varphi\;\cong\;\varphi(R)\]
More practice problems: first isomorphism theorem for rings
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