# Tagged Questions

This tag is for questions about rings, which are a type of algebraic structure studied in abstract algebra and algebraic number theory.

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### Showing Quotient ring is a field using maximal Ideal

Question: Show that $R\left [ x \right ]/\left \langle x^{2}+1 \right \rangle$ is a field. Recall: Theorem: Let R be a commutative ring R with unity. Let I be a proper Ideal of a ring R. ...
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### Verify size of factor ring

Let the ring $R=\left \{ \begin{bmatrix} a_{1} &a_{2} \\ a_{3}& a_{4} \end{bmatrix} \mid a_{i} \in \mathbb{Z} \right \}$ and let I be the subset of R consisting of matrices with even ...
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### Is there a way to characterize the prime ideals in $\mathbb{R}[x_1,x_2, \dots , x_n]$?

I'm studying algebras which can be formed by the quotient of principal ideals in $\mathbb{R}[x_1, \dots , x_n]$, and thus would like to be able to determine which of said principal ideals are maximal, ...
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### A modularity condition

Let $R\subseteq S$ be rings with unity and $X$ ,$Y$ be subsets of $S$ with $X$ an ideal. If $S=X+Y$ what conditions should be held to infer the equality $R=(X∩R)+(Y∩R)$? I think that if we ...
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### Intersection of any set of ideals is an ideal

Prove that the intersection of any set of Ideals of a ring is an Ideal. I'm looking for hints. Let A, B both be Ideals of a ring R. Suppose $I \equiv A\cap B$. Since A and B are both Ideals of a ...
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### Showing <n> =nZ is a prime ideal of Z IFF n is a prime

Question: If n is an integer greater than 1, show that $\left \langle n \right \rangle=n\mathbb{Z}$ is a prime ideal of $\mathbb{Z}$ IFF n is a prime. I have a bit of problem proving the only ...
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### Proof Verification of Result Involving Maximal Ideals

In further investigation of a question I asked earlier, I came across the following result, the proof of which I hope can be looked over here. I personally find it kind of interesting and I hope ...
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### Visualising the kernel of a homomorphism and quotient groups

I am trying to represent myself quotient groups and I'm having trouble seeing what the kernel of a homomorphism : $\Phi: G \rightarrow G/H$ is (be it a ring homomorphism or a group homomorphism). I ...
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### How many complexes modulo a prime $p$ are of multiplicative order $p^2 - 1$?

If $i = \sqrt{(-1)} \bmod p$, $p$ prime, does not exist, then we can form numbers of the form $a+b i \bmod p$ with multiplicative order $p^2 - 1$. How often do these numbers occur modulo $p$? In ...
### How quickly can we detect if $\sqrt{(-1)}$ exists modulo a prime $p$?
How quickly can we detect if $\sqrt{(-1)}$ exists modulo a prime $p$? In other words, how quickly can we determine if a natural, $n$ exists where $n^2 \equiv -1 \bmod p$? NOTE This $n$ is ...