All Questions
Tagged with ideals solution-verification
194
questions
2
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1
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60
views
Unital Ring $R$ (Commutativity Not Assumed) is a Field if and only if Maximal Ideal is $0$.
I have only seen this statement proven under the assumption that $R$ is commutative. However, what if we dropped this assumption? (And before somebody comments this, I am aware that the ring will have ...
- 917
1
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0
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36
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Maximal m-system vs zero-divisors
Let $R$ be a commutative nilpotent-free ring with unity and $\mathfrak{M}$ be a maximal $m$-system. I want to show the following.
For any nonzero $a\notin \mathfrak{M}$; there exists $b\in \mathfrak{...
- 1,068
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0
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47
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$a$ is unit if and only if $a+I$ is unit.
Let R be a ring and I is ideal of it. We want to prove or disprove the statement.
Let $a$ be a unit. Then there is $a'$ such that $aa'=1$. The quotient ring R/I implies,
$(a+I)(a'+I)=aa'+I=1+I$ which ...
user1127460
1
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0
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44
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If $I\subseteq P_1\cup P_2$ with $P_1, P_2$ prime ideals of $R$, then $I\subseteq P_1$ or $I\subseteq P_2$.
Please note that I am not looking for a solution to the question in the title as it has been asked before.
I was trying to prove this famous result to a friend:
If $I$ is an ideal of a ring $R$ and $...
- 1,516
0
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1
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79
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Why my proof of every maximal ideal being prime is incorrect
I tried to prove that for a commutative ring $R$ with identity, every maximal ideal $I$ is prime. I think my proof was sort of going in the right direction, but this MSE answer was what I now realize ...
- 49
0
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1
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55
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Is the intersection of a homogenous ideal with a non homogenous ideal homogenous?
Let $A$ be a commutative graded ring, $I\subset A$ a homogenous ideal, and $J\subset A$ a non homogenous ideal. Is $I\cap J$ homogenous?
I feel like it should be. Indeed consider the following proof:
...
- 3,431
1
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2
answers
144
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Prove that $x^2+I=-1+I$
I have given an ideal $I=\{(x^2+1)\cdot p(x)|p(x)\in \mathbb{R}[x]\}$ which is an ideal of $\mathbb{R}[x]$. I have to prove that $x^2+I=-1+I$ holds in $\mathbb{R}[x]/I$.
I think, that I is also a ...
- 37
1
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0
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85
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Do we get a field when we form the quotient ring of polynomials over $\mathbb{Q}$ modulo polynomials that annihilate $\sqrt{2}+\sqrt{3}$?
Let $\alpha = \sqrt2+\sqrt3$. Prove that $I =\{f(x) \in \mathbb{Q}[x] : f(\alpha) = 0\}$ is a nontrivial principal ideal in $\mathbb{Q}[x]$. Discuss if $\mathbb{Q}[x]/I$ is a field.
This is my ...
- 275
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0
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39
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Using Finite Rank Operators to Prove that Compact Operators form an Ideal
This is from Analytic K-Homology by Higson and Roe.
Show that if $S$ is a rank one operator and $T$ is any operator, then $ST$ and $TS$ are rank one operators. Deduce that $\mathcal{K(H)}$ forms a ...
- 789
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2
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94
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understanding finitely generated ideals
Let $R$ be a commutative ring and $A \subset R$ be any ideal. The ideal $A$ is finitely generated if every $a \in A$ can be written in the form $a = r_1a_1+r_2a_2+ \cdot + r_na_n$ where $r_i \in R$ ...
- 1,328
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0
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101
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Let $R$ be a factorial ring in which every ideal generated by two elements is a principal ideal. Show that $R$ is a principal ideal ring. [duplicate]
I want to check if my solutions for this problem are right.
Let $R$ be a factorial ring in which every ideal generated by two elements is a principal ideal.
Show that $R$ is a principal ideal ring.
...
- 900
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0
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59
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Is this solution to show that the Ideal given by the kernel of$ f \in R \mapsto f(0,0) \in \mathbb C[X,Y]$ is not finitely generated correct?
Let $R \subseteq \mathbb C[X,Y]$ be the subring of all polynomials $f \in \mathbb C[X,Y]$ that can be written as $f = g(X)+X ·h(X,Y)$
a. Let $I \subset R$ be the kernel of the evaluation map $R \...
- 3,162
1
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1
answer
45
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Prove that $R/U$ forms a group under addition defined in $R.$ ($R$ and $U$ is a ring and ideal respectively)
If $R$ is a ring and $U$ be the (two-way) ideal of a ring. Then we define, $R/U$ to be the set of all cosets of $R$ under addition. Prove that $R/U$ forms a group under addition defined in $R.$
I ...
- 3,073
1
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0
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48
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Show that $\mathbb{Q}[x] \backslash \langle x^2-3 \rangle$ is isomorphic to $\mathbb{Q}(\sqrt{3})$ [duplicate]
I want to proof that show that $\mathbb{Q}[x] \backslash \langle x^2-3 \rangle$ is isomorphic to $\mathbb{Q}(\sqrt{3})$ here is my attempt using the fundamental theorem of homomorphisms.
Let's define $...
- 1,810
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1
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80
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Confusion about $\mathbb{Z}[x]$ being not a PID and generators of ideals of form $\left(f_1(x), \ldots, f_n(x)\right)$ [duplicate]
Not principal ideal. It's a well known fact, that $\mathbb{Z}[x]$ is not a PID, for example consider the following ideal
\begin{align}
I = \left(x, x + 2\right) = \{a(x)(x+2) + b(x)x| a(x), b(x) \in \...
- 927