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.

3,912 questions
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Finding associated primes of quotient modules

Consider the ideal $I=(a)\cap(a,b)^2=(a^2,ab)$ in $k[a,b]$ and set $R=k[a,b]/I$. The problem is to show that for all $n\ge 1$ and all $\lambda\ne 0$, the ideals $(b^n)$ and $(a+\lambda b^n)$ of $R$ ...
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Proof of every non-unit belongs to some maximal ideal. [duplicate]

I want to prove that every non-unit belongs to some maximal ideal. I did the following. Consider a commutative ring $R$ with unity. Consider a maximal ideal $M$. Also consider an element $r\notin M$. ...
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A typo in Eisenbud's Theorem 3.10?

Let $R$ be a Noetherian ring and $M$ a f.g. $R$-module. Let $M'$ be a proper submodule of $M$ and let $M'=\cap_{i=1}^n M_i$ be a primary decomposition with $M_i$ a $P_i$-primary submodule. Part (c) ...
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Does a ring homomorphism $\phi: R \rightarrow S$ give rise to any map $\psi: R/I \rightarrow S/J?$

Let $R, S$ be rings. Suppose $\phi: R \rightarrow S$ is a ring homomorphism. Clearly we have a map $R \rightarrow S/J$ defined by the canonical map. However, for any ideal $I \subset R,$ can I ...
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There exists a prime ideal not containing non-nilpotent element [duplicate]

I've been trying to solve the following problem. Let $A$ be a commutative ring with identity and $a \in A$ a non-nilpotent element, i.e., $a^m \neq 0$ for all $m \in \mathbb{Z}^+$. Prove there exists ...
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Localization of the polynomial ring at a prime ideal modulo maximal ideal is isomorphic to polynomial ring modulo prime ideal.

Let $p \in K[T]$ irreducible, s.t. $\text{LC}(p) = 1$. Then $$K[T]/(p) \cong K[T]_{(p)}/pK[T]_{(p)}.$$ What I have is: \begin{align*} &K[T] \hookrightarrow K[T]_{(p)} \text{ and } K[T]_{(p)} \...
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If $R$ is an n-fir, why free $R$-modules of rank at most $n$ have unique rank?

I am studying the Cohn's book "Free ideal Rings and Localization in General Rings", and there is something he takes for granted and I cannot find its reason in any part of the book. The question is: ...
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Is $\{ (\alpha ^ 2, \alpha) | \alpha \in \mathbb{R}\} \subseteq \mathbb{C}^2$ an algebraic variety?

Is $S = \{ (\alpha ^ 2, \alpha) | \alpha \in \mathbb{R}\} \subseteq \mathbb{C}^2$ an algebraic variety? I think the answer is no, and this was my approach: If $S$ is an algebraic variety then $I(S)$...
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Is this ideal principal?

Let $I = \{ f(x)∈ \mathbb{Z}[x] | f(0) \text{ is an even integer}\}$ This is an ideal of the ring $R = \mathbb{Z}[x]$. Is it principal? I have the definition of a principal ideal but I'm unsure if ...
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Ideal of $\mathbb{Z}[x]$ generated by $3$ and $x^2+1$ is proper

Let $I$ be the ideal of $\mathbb{Z}[x]$ generated by $3$ and $x^2+1$. Show that 1) $I$ is proper, 2) $\phi_a(I)=\mathbb{Z}$ for all $a\in\mathbb{Z}$, where $\phi_a(I):=\{f(a)\mid f\in I\}$ (...
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Ideal generated by two elements is maximal in $\mathbb{Z}[x]$

The question is to show that $I = (x^4 + x^3 + x^2 + x + 1, 2) \subset \mathbb{Z}[x]$ is a maximal ideal. I'm familiar with the results that $R / I$ is a field iff $I$ is maximal, and $R/I$ is a ...
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How to prove that $\dim_k(k[x_1,\dotsc,x_n]/\sqrt{I})=|V(I)|$?

Let $k$ be an algebraically closed field. How to prove that $\dim_k(k[x_1,\dotsc,x_n]/\sqrt{I})=|V(I)|$? I know that Hilbert Nullstellensatz will be required, but I can't figure out how. With the ...
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Find a homomorphism that maps a point in a Boolean algebra into the image of a proper filter

Let $\mathbb B$ be a Boolean algebra. Let $F$ be a proper filter in $\mathbb B$ (i.e. $0 \notin F$), and let $I$ be its dual ideal. Suppose there exists $a \in \mathbb B$ such that $a \notin F \cup I$....
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Does the vector space spanned by a Gröbner basis depend on the monomial order?

Let $k$ be a field, $n$ a natural number, and $I$ an ideal of $k[x_0,\dots,x_{n-1}]$. Given a monomial order $M$ on $k[x_0,\dots,x_{n-1}]$ let $G_M$ be the Gröbner basis of $I$ with respect to $M$ and ...
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Polynomial Ring modulo Ideal is Polynomial ring of cosets of indeterminates.

I wonder if for any arbitrary ideal $I \leq K[X_1,\ldots,X_n]$, the following is true: $$K[X_1,\ldots,X_n] \text{ mod } I = K[X_1 + I, \ldots, X_n +I].$$ If so, how can one show that?
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On monomial ideals and ring generated by monomials

Question 1: Is $(x^4,x^3y,x^2y^2,xy^3,y^4)$ a maximal ideal in $\mathbb C [ x^4,x^3y,x^2y^2,xy^3,y^4]$? Question 2: Are the ideals $(x^4,x^3y,x^2y^2,xy^3,y^4)$ and $(x^4,x^3y,xy^3,y^4)$ distinct in ...
Let $B$ be a Boolean algebra and $S \subseteq B$ be a subset that generates $B$. Is it the case that every filter $x$ of $B$ is equal to the filter generated by $x \cap S$? What if $S$ itself is a ...
Question (1): let $J$ be an ideal in $\mathbb C[X,Y]$ such that $J^2=(X,Y)^2$, then is it true that $J=(X,Y)$ ? Question (2): let $J$ be an ideal in $\mathbb C[X,Y,Z]$ such that $J^2=(X,Y,Z)^2$, then ...