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.

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 ?

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? […]

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

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

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?

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?

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 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) […]

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 […]

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