## Every Prime Ideal of a Finite Commutative Ring is Maximal

## Problem 723

Let $R$ be a finite commutative ring with identity $1$. Prove that every prime ideal of $R$ is a maximal ideal of $R$.

Add to solve laterLet $R$ be a finite commutative ring with identity $1$. Prove that every prime ideal of $R$ is a maximal ideal of $R$.

Add to solve laterLet $R$ be a commutative ring with $1$.

Suppose that the localization $R_{\mathfrak{p}}$ is a Noetherian ring for every prime ideal $\mathfrak{p}$ of $R$.

Is it true that $A$ is also a Noetherian ring?

Let $R$ be a commutative ring with $1$.

Prove that if every proper ideal of $R$ is a prime ideal, then $R$ is a field.

Add to solve later**(a)** Prove that every prime ideal of a Principal Ideal Domain (PID) is a maximal ideal.

**(b)** Prove that a quotient ring of a PID by a prime ideal is a PID.

**(a)** Let $F$ be a field. Show that $F$ does not have a nonzero zero divisor.

**(b)** Let $R$ and $S$ be nonzero rings with identities.

Prove that the direct product $R\times S$ cannot be a field.

Let $R$ be a commutative ring with identity $1\neq 0$. Suppose that for each element $a\in R$, there exists an integer $n > 1$ depending on $a$.

Then prove that every prime ideal is a maximal ideal.

Add to solve laterProve that any field automorphism of the field of real numbers $\R$ must be the identity automorphism.

Add to solve later Let $R$ be a ring with $1$. Suppose that $R$ is an integral domain and an Artinian ring.

Prove that $R$ is a field.

Let $R$ be a ring with $1$. Prove that the following three statements are equivalent.

- The ring $R$ is a field.
- The only ideals of $R$ are $(0)$ and $R$.
- Let $S$ be any ring with $1$. Then any ring homomorphism $f:R \to S$ is injective.

Let $R$ be a commutative ring with unity.

Then show that every maximal ideal of $R$ is a prime ideal.

Let $R$ be the ring of all continuous functions on the interval $[0, 2]$.

Let $I$ be the subset of $R$ defined by

\[I:=\{ f(x) \in R \mid f(1)=0\}.\]

Then prove that $I$ is an ideal of the ring $R$.

Moreover, show that $I$ is maximal and determine $R/I$.

Consider the cubic polynomial $f(x)=x^3-x+1$ in $\Q[x]$.

Let $\alpha$ be any real root of $f(x)$.

Then prove that $\sqrt{2}$ can not be written as a linear combination of $1, \alpha, \alpha^2$ with coefficients in $\Q$.

Prove that the polynomial

\[f(x)=x^3+9x+6\]
is irreducible over the field of rational numbers $\Q$.

Let $\theta$ be a root of $f(x)$.

Then find the inverse of $1+\theta$ in the field $\Q(\theta)$.

Let $F$ be a field and let

\[H(F)=\left\{\, \begin{bmatrix}

1 & a & b \\

0 &1 &c \\

0 & 0 & 1

\end{bmatrix} \quad \middle| \quad \text{ for any} a,b,c\in F\, \right\}\]
be the **Heisenberg group** over $F$.

(The group operation of the Heisenberg group is matrix multiplication.)

Determine which matrices lie in the center of $H(F)$ and prove that the center $Z\big(H(F)\big)$ is isomorphic to the additive group $F$.

Add to solve laterLet $\Q$ be the field of rational numbers.

**(a)** Is the polynomial $f(x)=x^2-2$ separable over $\Q$?

**(b)** Find the Galois group of $f(x)$ over $\Q$.

Let $p\in \Z$ be a prime number and let $\F_p$ be the field of $p$ elements.

For any nonzero element $a\in \F_p$, prove that the polynomial

\[f(x)=x^p-x+a\]
is irreducible and separable over $F_p$.

(Dummit and Foote “Abstract Algebra” Section 13.5 Exercise #5 on p.551)

Add to solve laterShow that fields $\Q(\sqrt{2}+\sqrt{3})$ and $\Q(\sqrt{2}, \sqrt{3})$ are equal.

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Let $R$ be a commutative ring with $1$. Prove that the principal ideal $(x)$ generated by the element $x$ in the polynomial ring $R[x]$ is a prime ideal if and only if $R$ is an integral domain.

Prove also that the ideal $(x)$ is a maximal ideal if and only if $R$ is a field.

Add to solve laterLet $R$ be a ring with unit $1\neq 0$.

Prove that if $M$ is an ideal of $R$ such that $R/M$ is a field, then $M$ is a maximal ideal of $R$.

(Do not assume that the ring $R$ is commutative.)