Exercise · ZW-FIRST-ISOMORPHISM-THEOREM-FOR-RINGS-010 · ★★★ Advanced
Which of these functions \(\mathbb{Z}\times \mathbb{Z}\to \mathbb{Z}\) is a ring homomorphism with \(\varphi(1,1)=1\)?
Choose one option.
Show answer
- C. \(\varphi(a,b)=a\)
Explanation
The projection preserves sum and product. The function \(a+b\) fails on products: \((1,0)(0,1)=(0,0)\) but \(1\cdot1\ne0\); the difference and \(ab\) do not preserve the sum or the product in general.
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?
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- 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-006 There is a surjective homomorphism ℤ_30→ ℤ_5. How many elements does its kernel have?
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- 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?