If the Quotient Ring is a Field, then the Ideal is Maximal
Let $R$ be a ring with unit $1\neq 0$.
Prove that if $M$ is an ideal of $R$ such that $R/M$ is a field, then $M$ is a maximal ideal of $R$.
(Do not assume that the ring $R$ is commutative.)
Proof.
Let $I$ be an ideal of $R$ such that
\[M \subset I \subset […]

Find a Nonsingular Matrix Satisfying Some Relation
Determine whether there exists a nonsingular matrix $A$ if
\[A^2=AB+2A,\]
where $B$ is the following matrix.
If such a nonsingular matrix $A$ exists, find the inverse matrix $A^{-1}$.
(a) \[B=\begin{bmatrix}
-1 & 1 & -1 \\
0 &-1 &0 \\
1 & 2 & […]

Pullback Group of Two Group Homomorphisms into a Group
Let $G_1, G_1$, and $H$ be groups. Let $f_1: G_1 \to H$ and $f_2: G_2 \to H$ be group homomorphisms.
Define the subset $M$ of $G_1 \times G_2$ to be
\[M=\{(a_1, a_2) \in G_1\times G_2 \mid f_1(a_1)=f_2(a_2)\}.\]
Prove that $M$ is a subgroup of $G_1 \times G_2$.
[…]

The Rank and Nullity of a Linear Transformation from Vector Spaces of Matrices to Polynomials
Let $V$ be the vector space of $2 \times 2$ matrices with real entries, and $\mathrm{P}_3$ the vector space of real polynomials of degree 3 or less. Define the linear transformation $T : V \rightarrow \mathrm{P}_3$ by
\[T \left( \begin{bmatrix} a & b \\ c & d \end{bmatrix} \right) = […]

The Group of Rational Numbers is Not Finitely Generated
(a) Prove that the additive group $\Q=(\Q, +)$ of rational numbers is not finitely generated.
(b) Prove that the multiplicative group $\Q^*=(\Q\setminus\{0\}, \times)$ of nonzero rational numbers is not finitely generated.
Proof.
(a) Prove that the additive […]

If a Matrix is the Product of Two Matrices, is it Invertible?
(a) Let $A$ be a $6\times 6$ matrix and suppose that $A$ can be written as
\[A=BC,\]
where $B$ is a $6\times 5$ matrix and $C$ is a $5\times 6$ matrix.
Prove that the matrix $A$ cannot be invertible.
(b) Let $A$ be a $2\times 2$ matrix and suppose that $A$ can be […]

A Group of Order $20$ is Solvable
Prove that a group of order $20$ is solvable.
Hint.
Show that a group of order $20$ has a unique normal $5$-Sylow subgroup by Sylow's theorem.
See the post summary of Sylow’s Theorem to review Sylow's theorem.
Proof.
Let $G$ be a group of order $20$. The […]

Polynomial $(x-1)(x-2)\cdots (x-n)-1$ is Irreducible Over the Ring of Integers $\Z$
For each positive integer $n$, prove that the polynomial
\[(x-1)(x-2)\cdots (x-n)-1\]
is irreducible over the ring of integers $\Z$.
Proof.
Note that the given polynomial has degree $n$.
Suppose that the polynomial is reducible over $\Z$ and it decomposes as […]