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Worksheet Weekly Practice 8
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 credit 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.
True/False, Multiple Choice, & Fill-In. For these problems a justification is not required for credit, but it may be useful for your own understanding to include one. True/False problems should be marked
True if the statement is always true, and
False otherwise. Multiple choice problems may have more than one correct answer if that is indicated in the problem statement; be sure to select all that apply. Fill-in problems require a short answer such as a number, word, or phrase.
1.
Definition: Give the definition of an ideal in a ring
\(R\text{.}\)
2.
Fill-In: The characteristic of an integral domain is either
or
3.
Fill-In: A proper ideal
\(I\) is called
if
\(ab\in I\) implies
\(a\in I\) or
\(b\in I\text{.}\)
Short Response. Your responses to these questions should be complete solutions with justifications, as per the
Grading Specifications .
4. Find all units, zero-divisors, idempotents, and nilpotent elements in
\(\Z_3\times \Z_6\text{.}\)
5. In
\(\Z_7\text{,}\) give a reasonable interpretation for the expressions
\(1/2, -2/3, \sqrt{-3}, -1/6\text{.}\)
6. Find a subring of
\(\Z\times \Z\) that is not an ideal of
\(\Z\times \Z\text{.}\)
7. How many elements are in
\(\Z[i]/\langle 3+ i\rangle\text{?}\) Give reasons for your answer.
8. Let
\(R=\Z_8\times \Z_{30}\text{.}\) Find all maximal ideals of
\(R\) and for each maximal ideal
\(I\) identify the size of the field
\(R/I\text{.}\)
