# Normal-distribution

• 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 […]