Questions tagged [ideals]
An ideal is a subset of ring such that it is possible to make a quotient ring with respect to this subset. This is the most frequent use of the name ideal, but it is used in other areas of mathematics too: ideals in set theory and order theory (which are closely related), ideals in semigroups, ideals in Lie algebras.
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Jacobson Radical
Could you please help me solve this question?
Let I1, I2 be ideals of a ring R (not necessarily commutative) s.t.
I1 + I2 = R
I1∩I2 = J(R) (where J(R) is the Jacobson radical)
Show that if x2 is an ...
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Equality of two completions
I have the following question.
Suppose $R$ is Noetherian ring, $I$ is ideal in $R$ and $S$ is multiplicatively closed set. Let $(I^n\colon\langle S\rangle) = \varphi^{-1}(I^nS^{-1}R),$ where $\varphi\...
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Jacobson radical and invertible element
Let $I_1,I_2$ be ideals of a ring $R$ such that $I_1+I_2=R$ and their intersection is contained in $J(R)$ (the Jacobson radical of $R$). Show that if $x_2$ is an element of $I_2$ s.t. $x_2+I_1$ is ...
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What is the simplest non-principal ideal?
Let's restrict ourselves to commutative rings (not necessarily with unity).
Is there a simpler example of a non-principal ideal than $\langle a,x\rangle$ in $R[x]$, where $a\in R$ is not a unit (and ...
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Show that $(3,\sqrt [3]{11}+1)$ is a principal ideal in $\mathbb{Z}[\sqrt[3]{11}]$
I know how to verify $(3,\sqrt [3]{11}+1)=(\sqrt[3]{11}-2)$:
$(\sqrt[3]{11}-2)(\sqrt[3]{121}+2\sqrt[3]{11}+4)=11-8=3$ and then $\sqrt[3]{11}+1=\sqrt[3]{11}-2+3$
But if I don't know the answer, how can ...
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Differences in meaning for notations $\alpha_i$ versus $\alpha(i)$ and meaning of $\beta(ij)$ for denoting axioms in monomials
The following are partly taken from Malik and Sen's Fundamentals of Abstract Algebra
Background
First we note that we can reconstruct the monomial $x^\alpha=x_1^{\alpha_1}\cdots x_n^{\alpha_n}$ from $...
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How do we find a primary decomposition of an ideal?
Currently I'm reading about primary decomposition of ideals from Atiyah and Macdonald's Introduction to Commutative Algebra book. I've read all the theorems related to primary decomposition given in ...
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Why are prime ideals proper?
As children we all learn this erroneous definition of a prime number: “a number $n\in \Bbb N$ is prime iff it’s only divided by one and itself”. Well that’s fine until the teacher asked us for ...
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Isomorphism $\mathbb Z[\omega]/(1-\omega)^2\cong (\mathbb Z/(p))[x]/(1-X)^2$, $\omega$ is the $p-$th root of unity.
Im reading the following proof of Fermat's Last Theorem from Keith Conrad
https://kconrad.math.uconn.edu/blurbs/gradnumthy/fltreg.pdf
On page 5 he mentions that $\mathbb Z[\omega]/(1-\omega)^2\cong (\...
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Find the ideal class group of $\mathbb{Q}(\sqrt{-5})$ by using the factorization theorem
Let $K=\mathbb{Q}(\sqrt {-5})$. We have shown that $\mathcal{O}_K$ has the integral basis $1,\sqrt{-5}$ and $D=4d=-20$. By computing the Minkowski's constant:$$M_K=\sqrt{|D|}\Big(\frac{4}{\pi}\Big)^{...
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If the quotient of an ideal is principal, is the original ideal principal?
Let $R$ be a unity conmutative ring and $I \subset J \subset R$ ideals of $R$. Is it true that if $J / I$ is principal, then $J$ is principal? This question has came to me on other excercise in which, ...
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Consider in $Z[x]$ the polynomial $f(x)=x^2+x+1$. How do I determine whether $I=(2,f(x))$ is a prime ideal?
Consider in $Z[x]$ the polynomial $f(x)=x^2+x+1$. How do I determine whether $I=(2,f(x))$ is a prime ideal? I read that if a polynomial is primitive then the generated ideal is prime (Why?). So $(f(x))...
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Are the ideals of a ring with cyclic additive group always principal?
Note for me rings need not be unital or commutative.
Let $R$ be a ring with cyclic additive group $(R, +, 0)$ and let $I$ be an ideal in $R$. Is $I$ principal?
Here's my attempt, assuming $R$ has a $...
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Question about an example on ring theory from Dummit and Foote
Background
Example: If $p$ is a prime, the ring $\Bbb{Z}[x]/p\Bbb{Z}[x]$ obtained by reducing $\Bbb{Z}[x]$ modulo the prime ideal $(p)$ is a Principal Ideal Domain, since the coeffiencets lie in the ...
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the order of $R=\Bbb{Z}[x]/(ax+b, x^2+5)$ is $5a^2+b^2$
Let $a,b \in \Bbb{Z}$. When $a\neq 0$, I want to prove the order of $R=\Bbb{Z}[x]/(ax+b, x^2+5)$ is $5a^2+b^2$.
$R\cong \Bbb{Z}[-b/a]/((-b/a)^2+5)$. If I could prove the last ring is isomorphic to $\...
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