Exercise · ZW-FIRST-ISOMORPHISM-THEOREM-FOR-RINGS-001 · ★☆☆ Basic
Let \(\varphi\colon \mathbb{Z}\to \mathbb{Z}_{7}\) be reduction modulo 7. What is \(\ker\varphi\)?
Choose one option.
Show answer
- B. \(7\mathbb{Z}\)
Explanation
The kernel consists of the integers whose residue modulo 7 is 0, that is, the multiples of 7. It is an ideal of \(\mathbb{Z}\), not a subset of \(\mathbb{Z}_{7}\).
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 \(\varphi\).
\[R/\ker\varphi\;\cong\;\varphi(R)\]
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