Examples of Prime Ideals in Commutative Rings that are Not Maximal Ideals
Problem 520
Give an example of a commutative ring $R$ and a prime ideal $I$ of $R$ that is not a maximal ideal of $R$.
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Add to solve later Example of Two Groups and a Subgroup of the Direct Product that is Not of the Form of Direct Product
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.
Add to solve later Every Complex Matrix Can Be Written as $A=B+iC$, where $B, C$ are Hermitian Matrices
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.
Add to solve later Example of a Nilpotent Matrix $A$ such that $A^2\neq O$ but $A^3=O$.
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.)
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