# Tagged Questions

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, ...

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### Product of ideal generators

Can we in general say that if we have an ideal $(I,J)$ that this is the same as the ideal $(I,J,IJ)$, where $IJ$ is the product of I and J?
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### Ideals of $\mathbb{Z}[i]$ geometrically

It is pretty easy to visualize the ideals of $\mathbb{Z}$ in the "integer line". Let's go up to $\mathbb{Z}[i]$ and consider the ideal $3\cdot\mathbb{Z}[i]$. We can visualize it as a "sub-lattice" ...
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### Ideals in Lie algebras

Is it true that in a Lie algebra $\mathcal {L}$ the product of two ideals $[I, J]$ is equal to the intersection $I\cap J$?
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### how to do isomorphic ideal from root system of degree 5 or more [closed]

update1 i notice solving 1+x+x^2+x^3+x^4+x^5 have 5 solutions two conjugate real number and each of them having conjugate complex number part i change a*b + c to a1*a2*b1*b2 + c1*c2** still can ...
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### A example of a commutative chain ring [duplicate]

We say a commutative ring $R$ is a chain ring whenever its ideals form a chain with respect to inclusion. I am looking for a chain ring with Krull dimension two? Thank you for any help.
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### Find the order of the quotient ring $\mathbb Z[x]/J$

Let $J=\{f(x)\in \mathbb Z[x]:6\mid f(0)\}$. Show that $J$ is an ideal of $\mathbb Z[x]$, but not a prime ideal of $\mathbb Z[x]$. Also find the order of the quotient ring $\mathbb Z[x]/J$. I know ...
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### Is the Jacobson radical of a ring with finite spectrum and nilpotent nilradical nilpotent?

I tried to solve 1.3.3 in Bosch, Algebraic Geometry and Commutative Algebra. I did not find a way to solve it. But I found this: Finitely many prime ideals ⇒ cartesian product of local rings. And ...
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### Maximal ideals of polynomial ring $R[x]$ over a commutative ring $R$

In the article 1, Page 46, the author notes that: At this point it should be noted that there is an obvious connection with commutative rings. In fact, if $R$ is a commutative domain and $T= C$ ...
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### Non-invertible elements form an ideal

Problem is this: Suppose for any element $r$ in a ring $R$ with unity $1$, $r$ or $1-r$ is invertible. Then show that non-invertible elements form an ideal of $R$. It is crystal clear that $-r$ ...
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### A problem on maximal ideal in polynomial ring.

Let $\Bbb R[x]$ be the polynomial ring over $\Bbb R$ in one variable. Let $I\subseteq\Bbb R[x]$ be an ideal. Then which are true? $I$ is a maximal ideal if and only if $I$ is a non-zero prime ...
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### Number of ideals in a finite-dimensional $K$-algebra

Let $K$ be a field and $A$ be a finite-dimensional $K$-algebra. Does $A$ have finitely many ideals? (I know that $A$ has finitely many prime ideals.)
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### Intersection of a prime ideal with a non-prime ideal

Let $I_1$ be a prime ideal of $\mathbb{C}[X_1,...,X_n]$ and $I_2$ be another ideal of $\mathbb{C}[X_1,...,X_n]$ which is not a prime ideal. Assume further than $I_2 \not\subset I_1$. Under what ...
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### Finding Ideals in $\begin{bmatrix} \mathbb{Q} & \mathbb{Q}\\ 0 & 0 \end{bmatrix}$

I am looking to find the left, right and two sided ideals of the ring R = $\begin{bmatrix} \mathbb{Q} & \mathbb{Q}\\ 0 & 0 \end{bmatrix}$. It is in finding the left ideals that I am stuck. ...
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### The form of an element in $R[x]/I$ where $I$ is a principal ideal

It was stated in this question (Give an intuitive explanation for polynomial quotient ring, or polynomial ring mod kernel) that given a field $F$, a general element of $F[x]/(a_n x^n + \ldots + a_0)$ ...
If I were to visualize the ideal $(2, 3+3i)$ of $Z[i]$ on the complex plane, I would find a gcd of $2$ and $3+3i$ (for example $1+i$) and the ideal $(2, 3+3i)$ is identical to $(1+i)Z[i]$, which forms ...
The exercise asks us to prove that $I = \{ f \in \Bbb R[X,Y,Z] \mid f(a,b,c) = 0, ~\forall\,(a,b,c)\in \Bbb S^2 \}$ is a finitely generated ideal of $\Bbb R[X,Y,Z]$. Well, clearly $I$ is an ideal of \$...