# Tagged Questions

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### Algebraic Geometry and Maximal ideals

I am solving the following problem but couldn't figure out a strategy to solve: Does $(x^3-17, y^2)$ generate maximal ideals in the quotient ring $R=\mathbb{C}[x,y]/I$ where $I$ is the principal ...
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### Dfference between strongly prime and prime ideal

An ideal $P\subset R$ is strongly prime, if for any $x$ and $y$ in the quotient field of $R$, $xy\in P$ implies $x\in P$ or $y\in P$. What is the difference between strongly prime ideal of $R$ and a ...
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### The sum of all right ideals isomorphic as modules to a simple module is an ideal

I could use some help on the following problem. Let R be a ring. (a) If $r \in R$ and $U$ is a minimal right ideal of $R$, show that either $rU=0$, or that $rU$ and $U$ are isomorphic right ...
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### Real analysis based on rings and ideals [duplicate]

Let $R$ be the ring of all the real valued continuous functions on the closed unit interval. Show that $M=\{ f\in R:f(1/3)=0 \}$ is a maximal ideal
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### Checking if $\langle 2 \rangle$ is a maximal ideal in $\mathbb{Z}[i]$

Is $\langle 2\rangle$ a maximal ideal in $\mathbb Z[i]$? Solution: We know that $\mathbb Z[i]$ is ED And hence PID. Consider $2\in\mathbb Z[i]$. Then $N(2)=2^2=4$ (NOTE: $N$ is norm). Since $N(2)$ is ...
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### $IJ =(I\cap J)(I+J)$ holds in a PID? $I,J$ Ideals of a PID

One inequality is obvious, but the other one im not sure if holds. Any idea?
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### What's the difference between $B$ being integral over $A$ and $B$ being an integral extension of $A$?

I have the definitions: "$C=\{ b\in B:b\:\text{is integral over } A \}$ is called the integral closure of $A$ in $B$ If $C=B$ we say $B$ is integral over $A$." "$B$ is an integral extension of $A$ if ...
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### Common divisors in a PID

Suppose that R is an integral domain and that α, β, γ ∈ R. We say that γ is a common divisor of α and β if γ|α in R and γ|β in R. Suppose that R is a PID. Suppose that α, β ∈ R. Let I = (α) and ...
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### A quotient $\mathcal{O}/\mathfrak{a}$ of a Dedekind domain is principal (Neukirch exer 1.3.5)

The exercise states: The quotient ring $\mathcal{O}/\mathfrak{a}$ of a Dedekind domain by an ideal $\mathfrak{a}\ne 0$ is a principal ideal domain. The proof by localization ...
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### Why does the only maximal of $k[[X_1,\ldots,X_n]]$ is $(X_1,\ldots,X_n)$?

I'm trying to understand in this book why the only maximal of $k[[X_1,\ldots,X_n]]$ ($k$ field) is $(X_1,\ldots,X_n)$: If I prove $rad(k[[X_1,\ldots,X_n]])\subset (X_1,\ldots,X_n)$, (where $rad$ is ...
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### Prove: The pre-image of an ideal is an ideal.

Let $\phi : R \to S$ be a homomorphism. If $N$ is an ideal of $S$, then $\phi ^{-1} (N)$ is an ideal of $R$.
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### Principal prime ideal is generated by irreducible element

$R$ is an integral domain, $x\in R$ and $(x)=I$ is a prime ideal. Prove that $x$ is an irreducible element of $R$. So I assume $ab\in I$, with $a, b \in R$. Since $I$ is a prime ideal, either $a$ ...
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### Can someone explain ideals to me?

I'm struggling with the idea of ideals (both the definitions and the common notation). I'm in a basic collegiate algebra course, just looking for a bit of help. As simply defined as possible, if you ...
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### Maximal Ideals of $\mathbb{R}^{\infty}$

In the ring $\mathbb{R}^{\infty}$ (with the standard operations of component-wise addition and multiplication), what are the maximal ideals? It was quite simple to determine that the ideals with a ...
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### No units in quotient ring equivalent to no units in original ring?

Definition: Let $R$ be a ring with $1$. $r\in R$ is a unit if and only if $r \neq 1$ and there exists $s\in R, s \neq 1$ such that $rs=1=sr$. Let $R$ be a ring with $1$ and let $I$ be a proper ideal ...
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### What does $J_1\cap J_2=\emptyset$ mean algebraically for two varieties in $\Bbb{C}^n$?

Let $J_1, J_2$ be two varieties in ${\Bbb C}^n$. Then $$J_i=V(I_i)\quad i=1,2.$$ for some $I_i\subset\Bbb{C}[x_1,\cdots, x_n]$ and $$J_1\cap J_2=V(I_1\cup I_2)$$ and $$J_1\cup J_2=V(I_1I_2).$$ ...
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### What does it meet for an ideal to meet as set?

I'm trying to do a problem involving multiplicatively closed sets in a ring $R$ but I'm stuck as I don't know/understand what it means for an ideal to "meet" a multiplicatively closed set.
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### Unique Maximal ideal in a ring containing $\Bbb C$ is the nilradical.

This is a question from Algebra (Artin) ex.10.8.8 $R$ is a ring containing $\Bbb C$ as a subring. Assume $R$ is a finite dimensional vector space over $\Bbb C$ and that $R$ contains exactly one ...
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Let $R=k[x_1,\ldots,x_7]$ be a polynomial ring over field $k$ and $I=\bigcap_{i=1}^4 \mathfrak{p}_i$ where $\mathfrak{p}_1=(x_1,x_3,x_5,x_6), \mathfrak{p}_2=(x_1,x_3,x_4,x_6), ... 2answers 45 views ### Computing kernel Let$I,J$be two ideals of Noetherian ring$R$. How to compute kernel of following homomorphism directly: $$\phi: R/I\oplus R/J\to R/(I+J)$$ $$(a+I,b+J)\to (a-b)+I+J$$ 2answers 46 views ### Maximal Ideals in the Integers We know that the maximal ideals in$\mathbb{Z}$are all ideals of the form$(p)$, where$p$is prime. But what if we consider$(p, p_2)$, where$p_2$is prime, although I'm not sure that the ... 1answer 46 views ### How to prove that$J(M_n(R))=M_n(J(R))$? How to prove that$J(M_n(R))=M_n(J(R))$? Here$M_n(R)$is the ring of matrices of size$n^2$over the ring$R$. And$J(M_n(R))$is a two-sided ideal of the ring$M_n(R)$. 2answers 50 views ### Showing 3 is an irreducible element of$\mathbb Z[\sqrt{2}]$What I tried: $$(3)=\{3r|r\in \mathbb Z\}\space\mbox{is a maximal ideal of}\space\mathbb Z\implies(3)=\{3(a+b\sqrt{2})|a,b\in\mathbb Z\}\space\mbox{is a maximal ideal of }\mathbb ... 1answer 20 views ### In the ring R of real-valued functions on (0,1), I = \{f \in R \mid f(1/2)=0\text{ and }f(1/3)=0\} is not a prime ideal So I already proved that I is an ordinary ideal, now it's time to prove that it's not a prime ideal. So I need to show that R/P is not an integral domain, but I'm confused as to how to go about ... 1answer 39 views ### Clarification about some proof of Projectivity Small provides an example of a ring which is right but not left hereditary is the ring R =\left( \begin{matrix} \mathbb{Z} & \mathbb{Q} \\ 0 & \mathbb{Q} \end{matrix} \right); To ... 1answer 74 views ### Identifying some quotient rings How come that k[w,z]/(w^2+z,w^3 z^2)\cong k[w]/(w^7)? Also why is (xz,w)=(x,w)\cap(z,w) in the polynomial ring in 3 variables? what are the rules of ideal calculus making these results evident? 1answer 43 views ### Finish a proof that every prime ideal of a ring is the contraction of a prime ideal in its formal power series Given a commutative ring A with identity, and its formal power series ring A[[x]], I am attempting to prove that every prime ideal of A is the contraction of a prime ideal of A[[x]]. ... 2answers 34 views ### Finite generating set of the product of finitely generated ideals. Let R be a ring (not necessarilly unitary) and I=(a_1,\dots,a_n), J=(b_1,\dots,b_m) finitely generated ideals. By, definition$$ I = \bigcap \left\{\ I'\subseteq R\ \text{ideal}\ \big|\ a_i\in ... 2answers 65 views ### A question about proving maximal ideal Let$R$be the subring of the real numbers such that$R=\left\{a+b\sqrt{2}:a,b \in \Bbb Z\right\}$Let$M$be the ideal in$R$given by$M=\left\{a+b\sqrt{2}:\text{ a and b are divisible by ...
If $K$ is a field, it is easy to see that any non-zero ideal in the ring of formal power series $A=K[[X]]$ is of the form $AX^n$ with $n\geq 0$, so that $A=K[[X]]$ is a principal ideal ring. I ...