Articles of ring theory

Prove that ideals are prime

Prove that $\mathfrak{a}=\langle 3,4+\sqrt{-5}\rangle$ and $\mathfrak{b}=\langle7,4+\sqrt{-5}\rangle$ are prime ideals in ring $\mathbb{Z}[\sqrt{-5}]$. Any ideas? Thanks.

Subring of PID is also a PID?

Given a PID (Principal Ideal Domain), is every subring of PID also a PID ? Do I have to show that every subring of a PID is an ideal ?

Definition of a simple ring

I’m reading through Lang’s Algebra. Lang defines a simple ring as a semisimple ring that has only one isomorphism class of simple left ideals. On the other side, Wikipedia says that a simple ring is a non-zero ring that has no two-sided ideals except zero ideal and itself. I’m wondering, are these two definitions equivalent? […]

Proving that elements in a finite ring are zero divisors or units AND deducing that every finite integral is a field

This question already has an answer here: Every nonzero element in a finite ring is either a unit or a zero divisor 5 answers

Non-principal ideal in $K$?

This question already has an answer here: The ideal $I= \langle x,y \rangle\subset k[x,y]$ is not principal [closed] 6 answers

Example of a ring with prime characteristic, which is not an integral domain

We know that every intgral domain has prime (or 0) characteristic. Is there en example that the converse isn’t true? Does there exist a ring, which is not an integral domain, but has a prime characteristic?

$R_{a} = R/(x)$ isomorphic to $R_{b} = R/(x-1)$

I am looking at the following two rings: $R_{a} = R[x]/(x)$ and $R_{b} = R[x]/(x-1)$. I was told that these two rings were isomorphic, but I don’t see why. Is this due to the minimal polynomials?

Rings and Categories of Modules by Anderson and Fuller: Corollary $7.4$

I am reading the book Rings and Categories of Modules by Anderson and Fuller. I don’t understand corollary $7.4$ of that book. Can anyone explain that corollary to me? Thank for any insight.

Show that in a UFD, prime elements are irreducible

Show that in a UFD, prime elements are irreducible Here is what I have so far: Suppose $p \in R$ is prime, then the ideal it generates is a prime ideal in $R$. If $ab$ (where neither $a$ nor $b$ are zero) belongs to $\langle p \rangle$ then we can say (without loss of generality) […]

Field Extension Notation Indicative of Quotient Object

I’m pretty new to field theory and Galois theory and have been mildly puzzled by the notation $E/F$ to denote $E$ as a field extension of $F$. The notation seems reminiscent to me of quotient ring notation, $R/I$, to denote the ring formed by cosets of an ideal $I$. However, we can’t even form a […]