Tagged: example

Problem 520

Give an example of a commutative ring $R$ and a prime ideal $I$ of $R$ that is not a maximal ideal of $R$.

Problem 467

Give an example of two groups $G$ and $H$ and a subgroup $K$ of the direct product $G\times H$ such that $K$ cannot be written as $K=G_1\times H_1$, where $G_1$ and $H_1$ are subgroups of $G$ and $H$, respectively.

Problem 425

(a) Prove that each complex $n\times n$ matrix $A$ can be written as
$A=B+iC,$ where $B$ and $C$ are Hermitian matrices.

(b) Write the complex matrix
$A=\begin{bmatrix} i & 6\\ 2-i& 1+i \end{bmatrix}$ as a sum $A=B+iC$, where $B$ and $C$ are Hermitian matrices.

Problem 305

Find a nonzero $3\times 3$ matrix $A$ such that $A^2\neq O$ and $A^3=O$, where $O$ is the $3\times 3$ zero matrix.

(Such a matrix is an example of a nilpotent matrix. See the comment after the solution.)