## 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 laterProve that the quadratic integer ring $\Z[\sqrt{5}]$ is not a Unique Factorization Domain (UFD).

Add to solve laterProve that the quadratic integer ring $\Z[\sqrt{-5}]$ is not a Unique Factorization Domain (UFD).

Add to solve later Let $R$ be a commutative ring. Consider the polynomial ring $R[x,y]$ in two variables $x, y$.

Let $(x)$ be the principal ideal of $R[x,y]$ generated by $x$.

Prove that $R[x, y]/(x)$ is isomorphic to $R[y]$ as a ring.

Add to solve laterProve the following statements.

**(a)** If $a\neq 1$ is an idempotent element of $R$, then $a$ is a zero divisor.

**(b)** Suppose that $R$ is an integral domain. Determine all the idempotent elements of $R$.

It’s been a year since I started this math blog!!

More than 500 problems were posted during a year (July 19th 2016-July 19th 2017).

I made a list of the 10 math problems on this blog that have the most views.

Can you solve all of them?

The level of difficulty among the top 10 problems.

【★★★】 Difficult (Final Exam Level)

【★★☆】 Standard(Midterm Exam Level)

【★☆☆】 Easy (Homework Level)

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Prove that a positive definite matrix has a unique positive definite square root.

Add to solve later Let $A$ be a square matrix. A matrix $B$ satisfying $B^2=A$ is call a **square root** of $A$.

Find all the square roots of the matrix

\[A=\begin{bmatrix}

2 & 2\\

2& 2

\end{bmatrix}.\]

**(a)** Prove that the matrix $A=\begin{bmatrix}

0 & 1\\

0& 0

\end{bmatrix}$ does not have a square root.

Namely, show that there is no complex matrix $B$ such that $B^2=A$.

**(b)** Prove that the $2\times 2$ identity matrix $I$ has infinitely many distinct square root matrices.

Let $F$ be a finite field.

Prove that each element in the field $F$ is the sum of two squares in $F$.

Let $(\Q, +)$ be the additive group of rational numbers and let $(\Q_{ > 0}, \times)$ be the multiplicative group of positive rational numbers.

Prove that $(\Q, +)$ and $(\Q_{ > 0}, \times)$ are not isomorphic as groups.

Add to solve later Using the numbers appearing in

\[\pi=3.1415926535897932384626433832795028841971693993751058209749\dots\]
we construct the matrix \[A=\begin{bmatrix}

3 & 14 &1592& 65358\\

97932& 38462643& 38& 32\\

7950& 2& 8841& 9716\\

939937510& 5820& 974& 9

\end{bmatrix}.\]

Prove that the matrix $A$ is nonsingular.

Add to solve later Let $A$ be a square matrix.

Prove that the eigenvalues of the transpose $A^{\trans}$ are the same as the eigenvalues of $A$.

Prove that any field automorphism of the field of real numbers $\R$ must be the identity automorphism.

Add to solve laterLet $A$ be an $n\times n$ invertible matrix. Then prove the transpose $A^{\trans}$ is also invertible and that the inverse matrix of the transpose $A^{\trans}$ is the transpose of the inverse matrix $A^{-1}$.

Namely, show that

\[(A^{\trans})^{-1}=(A^{-1})^{\trans}.\]

Let $A$ be a singular $2\times 2$ matrix such that $\tr(A)\neq -1$ and let $I$ be the $2\times 2$ identity matrix.

Then prove that the inverse matrix of the matrix $I+A$ is given by the following formula:

\[(I+A)^{-1}=I-\frac{1}{1+\tr(A)}A.\]

Using the formula, calculate the inverse matrix of $\begin{bmatrix}

2 & 1\\

1& 2

\end{bmatrix}$.

Prove that if $A$ is a diagonalizable nilpotent matrix, then $A$ is the zero matrix $O$.

Add to solve later Prove that the ring of integers

\[\Z[\sqrt{2}]=\{a+b\sqrt{2} \mid a, b \in \Z\}\]
of the field $\Q(\sqrt{2})$ is a Euclidean Domain.

Find the inverse matrix of the $3\times 3$ matrix

\[A=\begin{bmatrix}

7 & 2 & -2 \\

-6 &-1 &2 \\

6 & 2 & -1

\end{bmatrix}\]
using the Cayley-Hamilton theorem.

Let $R$ be a ring with unit $1$. Suppose that the order of $R$ is $|R|=p^2$ for some prime number $p$.

Then prove that $R$ is a commutative ring.