# 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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### Is a ring R, modulo an ideal I (generated by x), then modulo an ideal J (generated by n) the same as R modulo the ideal generated by (n,x)?

Is the following statement true? $$R/(x,n) = \left[ R/(x) \right] / (n)$$ My thinking behind it was as follows: \begin{array}{ccc} \left[ R/(x) \right] / (n) & = & \{ r+(n) : r \in R/(x) \...
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### Invariant factors and the elementary divisors of the group $(\mathbb{Z}/77 \mathbb{Z})^{\times}$

If $A$ is a ring with unit element $1 \ne 0$ let $A^{\times}=\{a \in A: a$ invertible$\}$. Find the invariant factors and the elementary divisors of the group $(\mathbb{Z}/77 \mathbb{Z})^{\times}$. ...
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### Nilpotent element in $\mathbb{Z}/12\mathbb{Z}$ - Ideal

Let $A$ a commutative ring with the unit element $1 \not= 0$. $a \in A$ is a nilpotent element if there exists $n \in \mathbb{N}$ such that $a^n=0$. I have already prove that the set of nilpotent ...
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### Product of two principal ideals in $\Bbb Z[x]$

I'm looking for an easy argument for the following question: True or false, and why: The product of two principal ideals in $\Bbb Z[x]$ is a principal ideal. I know that $\Bbb Z[x]$ is not a ...
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### Show $Z(yf-1)$ is irreducible.

Question: $k$ is an algebraically closed field. Let $f \in k[x_1, \ldots, x_n]$ be an irreducible polynomial. Show that $Z(yf-1)\subseteq \textbf{A}^{n+1}$, with coordinates $x_1, \ldots, x_n, y$, ...
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### Each prime ideal contains an idempotent element

An element of ring $e$ is called idempotent iff $e^2=e$. Let $R$ be a commutative ring that contains the identity element and a non-trivial idempotent element. I want to show that each of ...
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### (0) is Maximal Ideal of Q(i) [closed]

Prove or disprove Ideal generated by $(0)$ is maximal ideal in $\mathbb{Q}(i)$.
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### Can every polynomial generate an ideal?

Suppose an arbitrary polynomial $f$ in a polynomial ring $R$. Is $\langle f\rangle$ always an ideal? Helper parts Consider a finite polynomial ring. Let $R=R[x_1,\ldots,x_n]$. Is the answer ...
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### How is “Binomial” defined in Algebraic Geometry?

I am learning ideal arithmetics and I was flabbergasted that $\langle x\rangle$ is binomial ideal, as observed with Macaulay2 here. $x$ is clearly not a polynomial with two terms. Then I read paper ...
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### Binomial ideals that are not toric i.e. binomial and not prime?

Let $R$ be a ring. Toric ideal $I$ is binomial (generated by a binomial) and prime (the quetient ring $R/I$ is integral domain). Paper and corollary 1.3 is to determine whether an ideal is binomial. ...
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### $R$ commutative with identity, $I$ finitely generated ideal. For each $r\in I$ there exists $m$ such that $r^mA\subseteq C$. Show $I^nA\subseteq C$.

I'm stuck on an exercise (E.VIII.4.1) from Algebra by Hungerford. It is stated as follows (sorry for the bad title; I didn't know how to fit it): Let $R$ be a commutative ring with identity and $I$...
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### Is the difference of ideals an ideal?

I am studying ideals and noticed that $I+J$ is an ideal as noted here. However the paper does not discuss $I-J$ so: Is the difference of ideals an ideal?
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### Has toric ideal something to do with torus?

I am studying ideals such as toric ideals but I am unable to find a consistent definition, it seems to be very general so please explain the origin of "toric ideal". Is there a geometric ...
Let $R$ be a ring and let $I\subseteq R$ the only maximal right ideal of $R$. I want to show that each element $a\in R-I$ is invertible. $I$ is also an ideal. Could you give me some hints what I ...