# Tagged Questions

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### Exercise related to commutative ring and finitely generated ideals

Let $R$ be a commutative ring with $1 \neq 0$. An ideal $I$ of $A$ is finitely generated if there are $r_1,...,r_n \in R$ such that $I=<r_1,...,r_n>$. Let $S$ be a multiplicative set of $R$. ...
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### Commutative ring and maximal ideal problem

Let $A$ be a commutative ring and $M$ be a proper maximal ideal in $A$. Show the following properties: (a) If each $a \in A \setminus M$ is a unit element in $A$, then $M$ is the only maximal ideal ...
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### Intersection of two polynomial ideals

In the 4-dimensional affine space $\mathbb{A}^4$ with coordinates $x,y,z,t$, consider $X$ as the union of the planes $$X'=\{x=y=0\}$$ and $$X''=\{z=x-t=0\}$$ (I'm working on a algebraically ...
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### Show that every maximal ideal in $\mathbb{Z}[x, y]$ contains a prime number [closed]

Let $\mathfrak{M} \subseteq \mathbb{Z}[x, y]$ be a maximal ideal. Show that $\exists\ p \in \mathbb{Z}$, $p$ prime such $p \in \mathfrak{M}.$ Thanks for the answers. I'd be interested in a proof ...
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### Find all the ideals of $\mathbb Q[X]$

I am trying to find all the ideals of the ring $\mathbb Q[X]$. If $I$ is a non trivial ideal of $\mathbb Q[X]$, then there exists $p(x) \in \mathbb Q[X]$. Since $I$ is an ideal and a group under ...
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### Left ideals of $M_n(K)$ [duplicate]

Let $K$ be a field and $n \in \mathbb N$. Show the following (i) Let $V \subset K^n$ be a subspace and $I_V$ the subset of $M_n(K)$ consisting of all the matrices whose rows belong to $V$. Prove that ...
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### Problem on the number of generators

I have got stuck with two generator problems: The ideal $(zx,xy,yz)$ can't be generated by $2$ elements The ideal $(xz-y^2,yz-x^3,z^2-xy)$ can't be generated by $2$ elements Here the ...
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### A finitely dimensional algebra over a field has only finitely many prime ideals all of them are maximal

Let $K$ be a field and let $R$ be a $K$-algebra with unity which is finite dimensional as a $K$-vector space. Prove that $R$ has only finitely many prime ideals all of which are maximal. (Hint: ...
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### Product of two non-principal ideals

I have problems understanding why $$(6,2+\sqrt{-56})(6,-2+\sqrt{-56})=6(2,\sqrt{-56})$$ in $\mathbb{Z}[\sqrt{-14}]$. By definition the product of two ideals $$IJ=\sum_{i,j}^{k}f_{i}g_{j}$$ ...
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### Analogy of ideals with Normal subgroups in groups.

I've started with Ideals in ring theory but still not comfortable with the analogy it has with normal subgroups in group theory.Like we can visualize normal subgroups as Is there some good intutive ...
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### Size of a subset of the set of units of a quotient ring

Let $R$ be a commutative Dedekind domain with multiplicative identity $1$, let $k$ be a positive integer, and let $I$ be a nonzero proper prime ideal of $R$. Is there a way to find the size of the set ...
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### Prove that $I$ is a maximal ideal of $\mathcal A$. [duplicate]

Please, give-me a hint to prove this proposition: Let $\mathcal A$ be the ring of all continuous real functions (with the usual operations of sum and multiplication) defined on the interval ...
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### In $\mathbb{Z}/(n)$, does $(a) = (b)$ imply that $a$ and $b$ are associates?

[Update: Based on the hints provided by @zcn and @whacka, I believe I have found a solution. See my answer below.] Below, $R$ is a commutative ring with $1$. In John J. Watkins' Topics in ...
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### When a given ideal is a radical ideal

I am wondering if there are any canonical methods for checking whether a given ideal is radical. For example, I got stuck on the following example: Let $f=x+2y-z$ and $g=z-2w$ and let $I$ and $J$ be ...
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### Characterizing Prime and Maximal Ideals in a nice Ring

Consider the "nice" ring $(\mathbb{Z}/20\mathbb{Z})[x]$ and I am trying to list all the prime and maximal ideals of this. The reason I call this a nice (or manageable) ring is because we ...
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### Nil radical of an ideal on a commutative ring

This is a problem of an exercise list: Let $J$ be an ideal of a commutative ring A. Show that $N(N(J))=N(J)$, where $N(J)=\{a \in A; a^n \in J$ for some $n \in \mathbb{N}\}$. What I did: ...
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### Polynomial Ideals and Transverse modules

I have a given ideal and I want to find the "smallest" ideal so that when I add it to the original one, I get another certain ideal. Let $\mathcal{E}$ be the ring of smooth function germs ...
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### Writing $I= (xz-y^2, yt- z^2)$ as an intersection of prime ideals

I need to write the ideal $I= (xz-y^2, yt- z^2) \subset R = \mathbb{K}[x,y,z,t]$ as intersection of prime ideals. Any idea? For the moment, I've noticed that $I$ is radical, then it suffices to ...
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### Irreducible ideals are prime in polynomial rings

Let $k$ be an algebraically closed field and $R$ the polynomial ring in $n$ variables over $k$. If $J$ is an irreducible ideal of $R$ then it is a prime ideal as well. To establish this statement ...