• Determine Whether Given Subsets in $\R^4$ are Subspaces or Not (a) Let $S$ be the subset of $\R^4$ consisting of vectors $\begin{bmatrix} x \\ y \\ z \\ w \end{bmatrix}$ satisfying $2x+4y+3z+7w+1=0.$ Determine whether $S$ is a subspace of $\R^4$. If so prove it. If not, explain why it is not a […]
• Non-Prime Ideal of Continuous Functions Let $R$ be the ring of all continuous functions on the interval $[0,1]$. Let $I$ be the set of functions $f(x)$ in $R$ such that $f(1/2)=f(1/3)=0$. Show that the set $I$ is an ideal of $R$ but is not a prime ideal.   Proof. We first show that $I$ is an ideal of […]
• Union of Two Subgroups is Not a Group Let $G$ be a group and let $H_1, H_2$ be subgroups of $G$ such that $H_1 \not \subset H_2$ and $H_2 \not \subset H_1$. (a) Prove that the union $H_1 \cup H_2$ is never a subgroup in $G$. (b) Prove that a group cannot be written as the union of two proper […]
• Every Group of Order 72 is Not a Simple Group Prove that every finite group of order $72$ is not a simple group. Definition. A group $G$ is said to be simple if the only normal subgroups of $G$ are the trivial group $\{e\}$ or $G$ itself. Hint. Let $G$ be a group of order $72$. Use the Sylow's theorem and determine […]
• Linearly Dependent Module Elements / Module Homomorphism and Linearly Independency (a) Let $R$ be a commutative ring. If we regard $R$ as a left $R$-module, then prove that any two distinct elements of the module $R$ are linearly dependent. (b) Let $f: M\to M'$ be a left $R$-module homomorphism. Let $\{x_1, \dots, x_n\}$ be a subset in $M$. Prove that if the set […]
• Dimension of the Sum of Two Subspaces Let $U$ and $V$ be finite dimensional subspaces in a vector space over a scalar field $K$. Then prove that $\dim(U+V) \leq \dim(U)+\dim(V).$   Definition (The sum of subspaces). Recall that the sum of subspaces $U$ and $V$ is \[U+V=\{\mathbf{x}+\mathbf{y} \mid […]
• If a Prime Ideal Contains No Nonzero Zero Divisors, then the Ring is an Integral Domain Let $R$ be a commutative ring. Suppose that $P$ is a prime ideal of $R$ containing no nonzero zero divisor. Then show that the ring $R$ is an integral domain.   Definitions: zero divisor, integral domain An element $a$ of a commutative ring $R$ is called a zero divisor […]