Instructions: You may type up or handwrite your work, but it must be neat, professional, and organized and it must be saved as a PDF file and uploaded to the appropriate Gradescope assignment. Use a scanner or scanning app to convert handwritten work on paper to PDF. I encourage you to type your work using the provided template.
All tasks below must have a complete solution that represents a good-faith attempt at being right to receive engagement credits. If your submission is complete and turned in on time, you will receive full engagement credits for the assignment. All other submissions will receive zero engagement credit. Read the guidelines at Grading Specifications carefully.
To abide by the class’ academic honesty policy, your work should represent your own understanding in your own words. If you work with other students, you must clearly indicate who you worked with in your submission. The same is true for using tools like generative AI although I strongly discourage you from using such tools since you need to build your own understanding here to do well on exams.
Suppose that \(R\) and \(S\) are commutative rings with identities, let \(\phi:R\to S\) be an onto ring homomorphism, and let \(I\) be an ideal of \(S\text{.}\)
Problem Specs/Notes: Be careful to show that all of the parts of the definition of prime and maximal ideals are satisfied. You do not need to show that the preimage of an ideal is an ideal, since we proved that already.
Problem Specs/Notes: You may either carefully construct an ideal properly containing \(\ideal{f}\) which is not all of \(\Z[x]\text{,}\) justifying both claims, or you may use the First Isomorphism theorem and argue that \(\Z[x]/\ideal{f}\) is not a field.
If \(R\) is a commutative ring and \(I\) and \(J\) are two proper ideals with \(I+J=R\text{,}\) prove that \(R/(I\cap J)\) is isomorphic to \(R/I\times R/J\text{.}\)
and \(a_n \neq 0\text{.}\) Prove that if \(r\) and \(s\) are relatively prime integers and \(f(r/s)=0\text{,}\) then \(r\mid a_0\) and \(s\mid a_n\text{.}\)
For a commutative ring with unity, we may define associates, irreducibles, and primes exactly as we did for integral domains. With these definitions, show the following.
Let \(n\) be a positive integer and \(p\) be a prime that divides \(n\text{.}\) Prove that \(p\) is irreducible in \(\Z_n\) if and only if \(p^2\) divides \(n\text{.}\)
An ideal \(A\) of a commutative ring \(R\) with identity 1 is said to be finitely generated if there exist elements \(a_1,a_2,\ldots, a_n\) of \(A\) such that \(A=\langle a_1,a_2,\ldots, a_n\rangle\text{.}\) An integral domain \(R\) is said to satisfy the ascending chain condition (ACC), if every strictly increasing chain of ideals \(I_1\subset I_2\subset \ldots\) must be finite in length. Show that an integral domain \(R\) satisfies the ascending chain condition if and only if every ideal of \(R\) is finitely generated.
