# Tagged Questions

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### How can we prove that every maximal ideal is a prime ideal? [on hold]

In abstract algebra,how can we prove that every maximal ideal is a prime ideal? Give full logical proof.
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### Colon ideal of fractional ideals is itself a fractional ideal

I received this question on homework in my homological algebra class and I need some guidance. Let $R$ be a commutative integral domain and $K$ be its field of fractions. A fractional ideal $I$ of ...
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### How to show that $\mathbb{C}[x_1,x_2,x_3, x_4]/(x_1x_2 - x_4x_3, x_1x_3 - x_2x_4, x_4x_1 - x_3x_2)$ is integral domain

I am looking for a way to show that the ring $\mathbb{C}[x_1,x_2,x_3, x_4]/I$ where $I = (x_1x_2 - x_4x_3, x_1x_3 - x_2x_4, x_4x_1 - x_3x_2)$ is an integral domain. In other words I want to show $I$ ...
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### Prove that, if $(a)=(a')$, then $a'=ua$

Let $R$ be integral domain. Show that if $2$ principal ideals $(a)$ and $(a')$ are equal (where $a,a'\in R$) then there exists $u\in R^{\times}$ such that, $a'=ua$ Now if $(a)=(a')$ then ...
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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 ...
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$\mathbb{R}[x]$ denotes the ring of polynomials in $x$ with real coefficients. Let $I \subset \mathbb{R}[x]$ be the subset of those polynomials with constant coefficient $0$, and let $J \subset ... 1answer 35 views ### Proving$M_p$is maximal in$C[0,1]$Let$M_p$be the ideal of those continuous functions of$C[0,1]$which have$p\in [0,1]$as a zero. It is a commonly known fact that$M_p$is a maximal ideal. However, the proof is generally ... 1answer 111 views ### Subset of$\mathbb{Z} \times \mathbb{Z}$I have a past exam question that is as follows: Let$k$be a fixed integer and$S = \{(a,ka)|a \in \mathbb{Z}\}$be a subset of$\mathbb{Z} \times \mathbb{Z}$. Prove that$S$is a subgroup of ... 1answer 54 views ### Every proper ideal$I$in a nonzero commutative unitary ring$R$is contained in a maximal ideal. If$R$is a nonzero commutative unitary ring, then$R$has a maximal ideal. Indeed, every proper ideal$I$in$R$is contained in a maximal ideal. There is a proof of this in Rotman's Advanced ... 0answers 24 views ### Need help with finding generator$I=\{a+bi \in R\mid a \equiv b\pmod 2\}$is ideal of$R=Z[i]=\{a+bi\mid a,b \in Z\}$. Can somebody help me to find the generator of$I\$?
Let $$I= \langle 11x^5y+7xy^6+9,8xy^4+6xy+9 \rangle$$ $$J= \langle 7x^5y^2+17x^2y^5+29,13xy^4+62xy^3+19 \rangle$$ ideals. Examine whether those two ideals are equal. By seeing their 3D plots I ...