Exercise · ZW-FIRST-ISOMORPHISM-THEOREM-FOR-RINGS-006 · ★★☆ Intermediate
There is a surjective homomorphism \(\mathbb{Z}_{30}\to \mathbb{Z}_{5}\). How many elements does its kernel have?
Choose one option.
Show answer
- C. \(6\)
Explanation
By the first theorem \(\mathbb{Z}_{30}/\ker\varphi\cong \mathbb{Z}_{5}\), so \(|\ker\varphi|=30/5=6\). The kernel is \(\{0,5,10,15,20,25\}\).
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
- ZW-FIRST-ISOMORPHISM-THEOREM-FOR-RINGS-001 Let φcolon ℤ→ ℤ_7 be reduction modulo 7. What is kerφ?
- ZW-FIRST-ISOMORPHISM-THEOREM-FOR-RINGS-002 Let πcolon ℤ× ℤ→ ℤ be given by π(a,b)=a. To which ring is (ℤ× ℤ)/kerπ isomorphic?
- ZW-FIRST-ISOMORPHISM-THEOREM-FOR-RINGS-003 How many elements does the kernel of the homomorphism φcolon ℤ_12→ ℤ_4, φ(a)=a bmod 4, have?
- ZW-FIRST-ISOMORPHISM-THEOREM-FOR-RINGS-004 Let φcolon ℤ_12→ ℤ_4 be reduction modulo 4, which is surjective. To which ring is ℤ_12/kerφ isomorp…
- ZW-FIRST-ISOMORPHISM-THEOREM-FOR-RINGS-005 If φcolon R→ S is any ring homomorphism, R/kerφ is isomorphic to:
- ZW-FIRST-ISOMORPHISM-THEOREM-FOR-RINGS-007 Let φcolon ℤ_6→ ℤ_2× ℤ_3 be given by φ(x)=(xbmod 2,\ xbmod 3). What is its kernel?
- ZW-FIRST-ISOMORPHISM-THEOREM-FOR-RINGS-008 Let φcolon ℤ→ ℤ_4× ℤ_6, φ(n)=(nbmod 4,\ nbmod 6). How many elements does the image of φ have?
- ZW-FIRST-ISOMORPHISM-THEOREM-FOR-RINGS-009 For which of these values of n is there a surjective ring homomorphism ℤ_12→ ℤ_n?
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