## 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 $R$.

Add to solve laterGive an example of a commutative ring $R$ and a prime ideal $I$ of $R$ that is not a maximal ideal of $R$.

Add to solve laterGive 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**(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.

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.)