Basic Ring Theory Exam
Sections from Dummit & Foote being tested on the midterm:
- §7.5 — 7.6
- §8.1 — 8.3
- §9.1 — 9.5
- §10.1 — 10.3
Here are the midterm and final exams for Math 542, Modern Algebra, at the University of Wisconsin-Madison in the spring semester 2015-2016, Professor Paul Terwilliger officiating. It is an undergraduate class, junior or senior level, for (mostly) math majors. The class takes up basic ring theory, following on Math 541, which is mostly group theory. The text, followed pretty closely, is Abstract Algebra, 3rd ed., by David S. Dummit and Richard M. Foote (Wiley, 2004) — D&F — widely used it appears. It certainly is comprehensive, with many examples and a great set of exercises, and is an impressive work in its own right, well-organized, demanding, and thorough. Typo-free as well, I haven't found a single one.
I'm publishing this thinking it might help math students bone up for their own tests. The midterm was given over fifty minutes with the idea that anyone knowing this stuff would have something reasonable to say about each problem in five minutes or so, if not providing a definitive solution for each one. I recommend proceeding as follows to get the most out of this:
1) Run through the problems quickly and pick one you've got a chance on. Spend some time on it (could be more than five minutes, studying for the test here, not taking it!).
2) If the problem is not yielding, hover the mouse over the word hintHint will appear here. next to the problem. A little tooltip should appear at the mouse with a hint.
3) If the hint is not helping, pore through D&F looking for the section relating to the problem. Study that section. A little opportunistically, sure, you want to solve the problem! But give it as much time as you can stand, trying to genuinely take in what is being written, the facts, the ramifications, the background. Note that a number of these problems are exercises in D&F.
4) If still not able to solve, click on the solution link for the problem to see a popup window with the solution. Even here, I'd say skim the solution in five seconds or less, trying to sense the key idea enabling you to solve it yourself.
5) If all else fails, read the solution carefully. Tie it to the relevant section of D&F, study the theorems and examples there, look for similar exercises at the end of that section, and try to conceive of similar problems. With any luck the question will exemplify the principle or theorem in your mind. Then you have made it yours and have truly succeeded, well on the way not only to passing your test, but to learning ring theory, the real goal.
The Midterm Exam
1) Let \( R \) and \( S \) denote nonzero rings. Prove that the ring \( R \times S\) is not a field (hintDoes \( R \times S\) have zero-divisors?, solution).
2) Prove that the polynomial ring \( \mathbb{Z}[x, y] \) is not a Euclidean domain (hintAre all ideals in \( \mathbb{Z}[x, y]\) principal?, solution).
3) Find all the ordered pairs \( r, s \) of positive integers such that \( r^2 + s^2 = 999 \) (hintLook at the prime power factorization., solution).
4) Let \( F \) denote a field and consider the polynomial ring \( R = F[x, y] \). Consider the ideals \( I = R(x - y^2) \) and \( J = R(x^2 - y^2) \) in \( R \). Prove that the quotient rings \( R / I \) and \( R / J \) are not isomorphic (hintOne of the polynomials is factorable, one is not., solution).
5) Let \( F \) denote a field. Let \( R \) denote the set of polynomials in \( F[x] \) that have \( x \)-coefficient zero. Note that \( R \) is a subring of \( F[x] \). Prove that \( R \) is not a UFD (hintConsider the factorization of \( x^6 \)., solution).
6) Prove that the polynomial \( x^3 + nx + 2 \) is irreducible in \( \mathbb{Z}[x] \), provided that \( n \neq 1, -3, -5 \) (hintUse the rational roots theorem., solution).
7) For the \( \mathbb{Z} \) modules \( M = \mathbb{Z} / 7 \mathbb{Z} \) and \( N = \mathbb{Z} / 6 \mathbb{Z} \), find all the elements in \( \text{Hom}_{\mathbb{Z}} (M, N) \) (hintgcd(7, 6) = 1., solution).
8) Let \( n = 1000. \) Find the order of the group of units for the ring \( \mathbb{Z} / n \mathbb{Z} \) (hintFactor 1000. , solution).
9) Let \( R \) denote a ring with \( 1 \neq 0 \). Let \( M \) denote an \( R \)-module, and let \( N \) denote an \( R \)-submodule of \( M. \) Define \( J = \{r \in R \; | \; ra = 0 \) for any \( a \in N \}. \) Prove that \( J \) is a 2-sided ideal in \( R \) (hintApply definitions. Easy!, solution).
10) Let \( R \) denote a commutative ring with \( 1 \neq 0. \) Let \( F \) denote a free \( R \)-module with finite rank. Prove that the \( R \)-modules \( \text{Hom}_R(F, R) \) and \( F \) are isomorphic (hintFor \( f \in F, \) define \( \varphi_f \) in terms of the basis representation of \( f. \), solution).
Additional sections from Dummit & Foote being tested on the final:
- §11.1 — 11.4
- §12.1 — 12.3
The final exam was in May 2016 and was comprehensive. Again ten questions, but now two hours for the exam. I found it more challenging than the midterm, probably because I'm weak in linear algebra and group theory, invariant factors and elementary divisors being featured on a couple of questions.
The Final Exam
1) Let \( x \) denote an indeterminate, and consider the ring of polynomials \( \mathbb{Z}[x] \) with integer coefficients. determine if the following polynomial is irreducible in \( \mathbb{Z}[x] \) (hintEasy — look at the divisors of the coefficients., solution):
\[ f(x) = x^6 + 30 x^5 - 15 x^3 +6x -120. \]
\[ f(x) = x^6 + 30 x^5 - 15 x^3 +6x -120. \]
2) Let \( F = \mathbb{Z} / 2 \mathbb{Z} \) denote the field with just two elements. Let \( x \) denote an indeterminate, and consider the ring of polynomials \( R = F[x]. \) Consider the ideal \( J \) of \( R \) generated by \( f(x) = x^4 + x^3 + x^2 + x + 1. \) Viewing the quotient ring \( R / J \) as a vector space over \( F, \) find the dimension and prove that your answer is correct (hintIs \( f(x) \) irreducible?, solution).
3) Let \( F \) denote a field, and consider a vector space \( V \) over \( F \) with dimension \( 5 \). Let \( T : V \rightarrow V \) denote a linear transformation whose Jordan canonical form consists of a single Jordan block with eigenvalue \( 0. \) (i) Find the dimension of the subspace \( U = \{v \in V \; | \; T^3 v = 0 \}. \) (ii) Find all subspaces \( W \) of \( V \) such that \( TW \subseteq W \) and the sum \( V = U + W \) is direct. Prove that your answer is correct (hintConsider \( T^3 \)'s action on the JCF basis., solution).
4) Consider the field \( \mathbb{R} \) of real numbers. Let \( x \) denote an indeterminate, and consider the vector space \( V \) over \( \mathbb{R} \) consisting of polynomials in \( x \) that have coefficients in \( \mathbb{R} \) and degree at most \( 3. \) Note that \( \{1, x, x^2, x^3\} \) is a basis for \( V. \) Consider the map \( \varphi : V \rightarrow \mathbb{R} \) that sends \( f \mapsto f(2) \) for all \( f \in V. \; \) (i) Prove that \( \varphi \) is in the dual space \( V^*. \; \) (ii) Express \( \varphi \) as a linear combination of the dual basis (hintEasy once you know what \( V^* \) and the dual basis are., solution).
5) Let \( R \) denote an integral domain. Let \( W \) denote a finitely generated \( R \)-module that is nonzero and torsion. Prove that the Annihilator of \( W \) is nonzero (hintFind a nonzero element annihilating all of \( W \)., solution).
6) Over the field of real numbers, find the rational canonical form of the matrix (hintFind \( c_A(x) \) and \( m_A(x). \), solution):
\[ A =
\begin{pmatrix}
-1 & -9 & 0 \\
1 & 5 & 0 \\
1 & 3 & 2
\end{pmatrix}.
\]
\[ A =
\begin{pmatrix}
-1 & -9 & 0 \\
1 & 5 & 0 \\
1 & 3 & 2
\end{pmatrix}.
\]
7) Consider the field \( F = \mathbb{Z} / 7 \mathbb{Z} \) of order \( 7.\) Consider the following matrix \( A \in \text{Mat}_7(F): \)
\[ A :
\begin{pmatrix}
0 & 0 & 0 & 0 & 0 & 0 & 1 \\
1 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 1 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 1 & 0
\end{pmatrix}. \]
Find the Jordan canonical form \( J \) for \( A. \) Prove that your answer is correct (hintConsider the effect of multiplication by \( A \)., solution).
\[ A :
\begin{pmatrix}
0 & 0 & 0 & 0 & 0 & 0 & 1 \\
1 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 1 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 1 & 0
\end{pmatrix}. \]
Find the Jordan canonical form \( J \) for \( A. \) Prove that your answer is correct (hintConsider the effect of multiplication by \( A \)., solution).
8) Recall the ring of integers \( \mathbb{Z} \) is a PID. Consider the \( \mathbb{Z} \)-module \( \mathbb{Z} / 100 \mathbb{Z}. \) (i) Find its invariant factor decomposition. (ii) Find its elementary divisor decomposition (hintFactor \( 100 \)., solution).
9) Let \( n = 1000. \) Let \( G \) denote the group of units for the ring \( \mathbb{Z} / n \mathbb{Z}. \) Find \( |G| \) (same as problem #8 on the midterm).
10) View the group \( G \) in Problem 9 as a \( \mathbb{Z} \)-module. For this module (i) find its invariant factor decomposition; (i) find its elementary divisor decomposition (hintFactor \( 1000 \) and break down \( \mathbb{Z}_{1000}^\times \)., solution).
Mike Bertrand
Madison, WI
May 20, 2016