1.
True/False: All rings are commutative.
Solution.
False. Many rings do not have commutative multiplication, such as any non-trivial matrix ring.
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Worksheet Weekly Practice 7
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.
2.
Fill-In: A is a commutative ring with identity and no zero divisors.
Solution.
This is the definition of an integral domain.
Short Response.
Your responses to these questions should be complete solutions with justifications, as per the Grading Specifications.
3.
Show that the set of Gaussian integers \(\Z[i]=\{a+bi \mid a,b \in \Z\}\) is a subring of the complex numbers.
Solution.
The Gaussian integers are a nonempty subset since all ordinary integers are also Gaussian integers. Now suppose \((a+bi),(c+di)\in \Z[i]\text{.}\) Then we have
\begin{equation*}
(a+bi)-(c+di)=(a-c)+(b-d)i
\end{equation*}
and
\begin{equation*}
(a+bi)(c+di)=(ac-bd)+(ad+bc)i,
\end{equation*}
and all of \(a-c,b-d,ac-bd,\) and \(ad+bc\) are integers when \(a,b,c,d\) are integers. So by the Subring Test \(\Z[i]\) is a subring of \(\C\text{.}\)
Problem Specs/Notes: This problem needs a complete application of the Subring Test for a Success.
4.
The ring \(\{0,2,4,6,8\}\) under addition and multiplication modulo 10 has an identity. Find it.
Solution.
Certainly an identity \(a\) in this ring must satisfy \(a^2=a\text{,}\) so we can rule out \(2,4,8\) immediately, as \(2^2=4, 4^2=6,8^2=4 \bmod 10\text{.}\) \(0\) cannot be the identity in a ring with more than one element, so it must be \(6\text{.}\) We can check this quickly also: \(6(2)=12=2, 6(4)=24=4, 6(6)=36=6, 6(8)=48=8\bmod 10\text{.}\)
Problem Specs/Notes: This problem needs complete justification of why your claimed element is the identity for a Success.
5.
Determine the smallest subring of \(\Q\) that contains \(1/2\text{.}\) That is, find the subring \(S\) with the property that \(S\) contains \(1/2\) and if \(T\) is any subring containing \(1/2\text{,}\) then \(T\) contains \(S\text{.}\)
Solution.
Let \(S\) be the desired subring. We claim that
\begin{equation*}
S=\left\{\frac{n}{2^k}, \mid n,k\in \Z, k\geq 0\right\}.
\end{equation*}
First, we show \(S\) is a subring. We have \(0\in S\text{,}\) so \(S\) is nonempty. Let \(n/2^k, m/2^\ell\) are in \(S\) and without loss of generality suppose \(k\geq \ell\text{.}\) Then we have
\begin{equation*}
\frac{n}{2^k}-\frac{m}{2^\ell}= \frac{n-2^{k-\ell}m}{2^k}
\end{equation*}
and \(n-2^{k-\ell}m \in \Z\) so \(S\) is closed under subtraction. We also have
\begin{equation*}
\frac{n}{2^k}\frac{m}{2^\ell}=\frac{nm}{2^{k+\ell}}
\end{equation*}
and \(nm, k+\ell \in \Z\) with \(k+\ell\geq 0\) so \(S\) is closed under multiplication. Hence \(S\) is a subring of \(\Q\text{.}\) Now suppose \(T\) is any subring containing \(1/2\text{.}\) By closure \(T\) contains all powers of \(1/2\) and all multiples of those powers. But these are exactly the elements of the form \(n/2^k\text{,}\) i.e. the elements of \(S\text{.}\) So \(T\) contains \(S\text{.}\) Hence \(S\) is the smallest subring of \(\Q\) that contains \(1/2\text{.}\)
Problem Specs/Notes: This problem needs a justification that your claimed set contains \(1/2\text{,}\) a proof that it is a subring of \(\Q\) and a proof that it is the smallest such subring for a Success.
