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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6310 • Algebra

Spring 2021 • Question 5

Question

Math 6310 • Algebra Spring 2021 • Question 5

Consider the ring homomorphism given by

\[ \begin {aligned} \mathbb {Z}[x]&\longrightarrow \mathbb {Z}\times \mathbb {Z}\\ f(x)&\longmapsto (f(1),f(-1)). \end {aligned} \]

Is this homomorphism surjective? Determine a minimal generating set for the kernel.

Raw LaTeX

Consider the ring homomorphism given by

\[
\begin{aligned}
\mathbb{Z}[x]&\longrightarrow \mathbb{Z}\times\mathbb{Z}\\
f(x)&\longmapsto (f(1),f(-1)).
\end{aligned}
\]


Is this homomorphism surjective? Determine a minimal generating set for the kernel.

Topic Tags

Hom Functors and Module Homomorphisms Polynomial Rings and Ring Homomorphism Kernels