# 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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### If $\phi:R\rightarrow R'$ is a surjective ring homomorphism and I is an ideal in R… continued below [duplicate]

then $\phi(I)=[s'\in R'|s=\phi(s)\space \forall\space s\in I]$ is an ideal in R' So I know that for something to be an ideal, it needs to be closed under subtraction and it must absorb products. I ...
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### Calculate generators of an intersection of Ideals

$$I = (x_1^2-x_1,x_2^2-x_2,...,x_n^2-x_n,t-\sum_{i=1}^n 2^{i-1}*x_i)$$ Ideal in $\mathbb Q[x_1, ..., x_n,t]$. How can I calculate the generators of $J = I \cap \mathbb Q[t]$ by hand? I tried it with ...
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### Prove: If $\mathit R$ is a commutative ring with unity and $\mathit I=(x)\subseteq R[x]$, then $R[x] / (x)\cong R$ [duplicate]

I know that to show a ring is isomorphic to another ring, I have to find a bijective ring homomorphism between the two rings. Or I could use the F.H.T. but I would also need a function to make that ...
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### Show that the product of ideals is equal to the intersection

I am following the notes of Gathmann to learn myself about commutative algebra. I have the following exercise written in them (without solutions at the end): Exercise 1.13. Show that the equation ...
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### Prove that every prime ideal that isn't maximal is a minimal prime ideal

Suppose that the additive group of the ring $R$ is a finitely generated abelian group. If $P$ is a maximal ideal of $R$, show that $R/P$ is a finite field. Show that every prime ideal of $R$ that is ...
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### Let J be an ideal. Find a function in I(V(J)) such that the function f is not in J

Let $J$ be the ideal $\langle x^2+y^2-1,y-1\rangle$. Find $f \in \textbf{I}(\textbf{V}(J))$ such that $f \not \in J$. I'm confused on a number of aspects here. Firstly, how do I find $\mathbf{V}(J)$ ...
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### Determining the ideals of a quotient ring

Given an ideal $I = \langle x^3 - x\rangle \subseteq \Bbb{R}[x]$, determine the ideals in the quotient ring $\Bbb{R}[x]/I$. I understand that the quotient ring is of the form $k[x_1...x_n]/I$ where ...
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### An ascending chain of prime ideals.

I am trying to get used to $\operatorname{Spec}$ of a ring. I know an example, when one prime ideal is contained in another for $\mathbb{C}[x,y]$. $(f) \subset (x-a,y-b)$, where $f(a,b) = 0$. Is ...
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### Find irreducible and connected components of $\operatorname{Spec}(\mathbb{C}[x] \times \mathbb{C}[y])$

Find irreducible and connected components of $\operatorname{Spec}(\mathbb{C}[x] \times \mathbb{C}[y])$ That's the problem 11.1 from commutative algebra course As answered here we can see the ...
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### When a two-generated ideal in $\mathbb{Z}[\sqrt{-3}]$ is the unit ideal or principal?

I am trying to figure out when the ideal $(a_1+b_1\sqrt{-3},a_2+b_2\sqrt{-3})$ is the unit ideal or principal in $\mathbb{Z}[\sqrt{-3}]$. Any hints?
Is commutator ideal compatible with direct sum? Let's take $\mathfrak{sl}_2(\Bbb K)\oplus\mathfrak{sl}_2(\Bbb K)$ which is semi-simple because $\mathfrak{sl}_2(\Bbb K)$ is simple Lie algebra. So we ...
Suppose $Q_1, Q_2\in \mathbb{C}[X_0,\dots,X_n]$ are irreducible homogeneous quadratic polynomials such that $V(Q_1, Q_2)$ is an irreducible projective variety of degree two and codimension two in \$\...