Articles of maximal ideals

Alternative proof of '$I$ is maximal iff $R/I$ is a field'

For any commutative ring $R$ and an ideal $I$ of $R$, $I \neq R$, show that $I$ is a maximal ideal iff $R/I$ is a field. I write my own proof and it checks with the ‘traditional’ proof which makes use of an element that is not in $I$. When I check the notes, my […]

Show that $M = \{a + b\sqrt{2}\mid 5\mid a, 5 \mid b\}$ is a maximal ideal in $R = \mathbb{Z}$ and find $R/M$.

I came across this question as part of my self-study of abstract algebra and so I would prefer answers suitable for a beginner. First I established that the set $$M = \{a + b\sqrt{2}\mid a, b\in \mathbb{Z}, 5\mid a, 5 \mid b\}$$ is an ideal in the ring $R = \mathbb{Z}[\sqrt{2}]$ and this is more […]

Let D be a principal ideal domain. Prove that every non-zero prime ideal in D is a maximal ideal in D.

Let D be a principal ideal domain. Prove that every non-zero prime ideal in D is a maximal ideal in D. So I’m think I need to use the fact that all PID’s are UFD’s. If it is a UFD I can infer that every irreducible element in D is prime. Does that mean it […]

Show that $I$ is an ideal

Let $R$ be a ring and $I\subseteq R$ the only maximal right ideal of $R$. I want to show that $I$ is an ideal. To show that $I$ is an ideal, we have to show that $I$ is a left ideal, right? How could we show that? Could you give me some hints?

prove the inverse image of a maximal ideal is also a maximal ideal

The following is the problem: Let $K$ be a field, and $A$ is a commutative ring containing $K$. $\phi:A \rightarrow K[X]$ is a ring homomorphism which is the identity on $K$. If $M$ is a maximal ideal of $K[X]$, show that $\phi^{-1}(M)$ is a maximal ideal of $A$. I tried to prove that $A/\phi^{-1}(M)$ is […]

Maximal ideals in polynomial rings

Let $K$ be a field. Let $\mathfrak{m}$ be an ideal of the polynomial ring $K[x_1,\ldots,x_n]$ and suppose the quotient $\frac{K[x_1,\ldots,x_n]}{\mathfrak{m}}$ to be isomorphic to $K$ itself. I want to prove that $\mathfrak{m}$ is of the form $$\mathfrak{m}=(x_1-a_1,\ldots,x_n-a_n)$$ for some $a_1,\ldots,a_n$ in $K$. All textbooks I consulted omit the proof of this fact, since it is […]

Proving that “Every non-trivial ring (i.e. with more than one element ) with unity has a maximal ideal” implies axiom of choice is true

I know that assuming axiom of choice or equivalently Zorn’s lemma , it can be proved that every non-trivial ring with unity has a maximal ideal (two sided ) . The wiki article on axiom of choice says that this statement regarding existence of maximal ideal in any non-trivial ring with unity is equivalent to […]

Examples of prime ideals that are not maximal

I would like to know of some examples of a prime ideal that is not maximal in some commutative ring with unity.

$\langle r\rangle$ maximal $\iff r$ irreducible

Let $R$ be a PID, and $r\in R- \{0\}$. Prove that $\langle r\rangle$ maximal $\iff r$ irreducible. “$\Leftarrow$”Easy. “$\Rightarrow$”If $J=\langle r \rangle$ then we will prove that $r$ is irreducible. If $r=ab$, we want to prove that $a\in U(R)$ or $b \in U(R)$. If we take the ideal which is generated by $\langle a\rangle$ then […]

Norm of powers of a maximal ideal (in residually finite rings)

Let $A$ be a residually finite integral domain and $M$ a maximal ideal in $A$. Is this true that $$|A/M^k|=|A/M|^k \quad (k\in\textbf{N}) \quad ?$$ In Hirano’s article On Residually Finite Rings we can read in page 11/14 (proof of proposition 4) an argument working in “Asano order”, and I have not the background to understand […]