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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Spectrum of a product of rings isomorphic to the product of the spectra

I've found in an exercise this statement: If $A$ is a commutative ring with unit and $A = A_{1} \times \dots \times A_{n}$ then $$\def\Spec{\operatorname{Spec}} \Spec(A) \cong \Spec(A_{1})\times ...
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34 views

Size of an ideal in a polynomial Ring

Let $F$ be a field and let $I = \{f(x) \in F[x]\mid f(a) = 0 ~~ \forall a \in F\}$. Prove that $I = \{0\}$ when $F$ is infinite. I have already shown that $I$ is an ideal and that $I$ is infinite ...
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1answer
137 views

Nilpotent Elements and Intersection of Prime Ideals

Prove that the set of nilpotent elements of a ring is the intersection of its prime ideals. I know these two useful facts: {nilpotent elements}$=\sqrt{0}$ $\sqrt{I}= \bigcap$ of prime ideals ...
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1answer
130 views

Ideals and prime ideals in a commutative ring. [closed]

Let $A_1$ and $A_2$ be two ideals, and $P_1$ and $P_2$ be two prime ideals in a commutative ring $R$. Assume that $A_1 ∩ A_2 ⊆ P_1 ∩ P_2$. Is there at least an $i$ and $j$ such that $A_i ⊆ P_j$ is ...
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50 views

Show an Ideal is the principal ideal for some polynomial.

Let $F$ be a field and $R = F[X]$. Suppose $I$ is an ideal of $R$. Show that $I = (p(X))$ for some $p(X)$ in $F[X]$. (Hint: consider a polynomial $p(X)$ of least degree in $I$.) I'm trying to do this ...
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1answer
72 views

generators of an ideal

I've been thinking about this exercise but I can't get the solution. In $\mathbb{R}^3$ , I consider the usual axis: $l_1=\{ x_1=x_2=0 \}$, $l_2=\{x_1=x_3=0\}$ and $l_3=\{ x_2=x_3=0 \}$. Calculate ...
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81 views

Problem about Gröbner basis.

I'd really appreciate if someone could help me. The problem is the following: If $\psi_1,...,\psi_m \in k[x_1,\dots,x_n]$ and consider the $k$-algebra homomorphism: ...
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1answer
57 views

Noetherian ring of Krull dimension $0$

I've found this claim: Let $A$ be a Noetherian ring of Krull dimension $0$ . Then $A$ is a field or it has a finite number of prime ideals. Why is this true ?
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68 views

prime ideals contains comaximal

Let $R$ be a commutative ring with unity 1 and $I$, $J$ and $P$ ideals in $R$ show that if every prime ideal of $R$ contains either $I$ or $J$ ,but not both then $I$ and $J$ are comaximal ...
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54 views

Proving the preimage of a ring homomorphism is not empty

I have a question from Abstract Algebra. Let $R$ and $S$ be commutative rings, and let $\phi:R \to S$ be a ring homomorphism. Show that if $J$ is an ideal of $S$, then $\phi^{-1}(J) := \lbrace r \in ...
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91 views

Vanishing polynomials

Let $K$ be a field and $V$ be the set of points $(t^3,t^4,t^5)$ where $t$ is in $K$. Set $I=(Y^2-XZ,Z^2-X^2Y,X^3-YZ)$. Show that $I$ is a subset of $A$, where $A$ is the set of polynomials which ...
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1answer
56 views

Degree of extension is equal to linear combination of prime factor multiplicities with prime factor index coefficients in Dedekind domains

I'm working on the following problem... Suppose that $A$ is a Dedekind domain with fraction field $K$. $L/K$ is a finite separable extension of $A$ of degree $n$, and $B$ is the integral closure ...
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1answer
204 views

Every radical ideal in a Noetherian ring is a finite intersection of primes

Prove that if $I$ is a radical ideal and $ab\in I$, then $I=rad(I+(a))\cap rad(I+(b))$. Deduce that every radical ideal in a Noetherian ring is a finite intersection of primes. I've done the ...
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146 views

Localisation and prime ideals

If $A$ is a ring and $S=\{1,f,f^2,f^3,...\}$ a multiplicative set of $A$. Prove that $Spec(A_f)=(\mathfrak{V}((f)))^c$. Notation: $A_f=S^{-1}A$ and $\mathfrak{V}((f))=\{P \in Spec(A): P \supset (f)\}$ ...
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111 views

How to show $\mathbb{Z}[x]/<2,x>$ is isomorphic to $Z_2$

I'm having quite a bit of trouble figuring out why $\mathbb{Z}[x]/<2,x>$ is isomorphic to $\mathbb{Z}_2$. So far I have figured out there is an onto map $\zeta: ...
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131 views

Irreducible components of $Spec(A) $

A topological space $X$ is called irreducible if given $A_{1}, A_{2} $ open sets $ \neq \emptyset $ then $A_{1} \cap A_{2} \neq \emptyset$. The maximal irreducible topological subspaces of $X$ are ...
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1answer
90 views

Primary descomposition of ideals

I'd appreciate if someone could help me a bit with this problem. Considering $\mathfrak{p}=(x,y), \mathfrak{q}=(x,z)$ and $\mathfrak{m}=(x,y,z)$ ideals in $k[x,y,z], k$ field. Is ...
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1answer
113 views

How to prove a subset is an ideal?

Consider the polynomial ring $GF (2)[x]/(x^n − 1)$. Let $f(x)$, $g(x)$ be two fixed polynomials in $GF(2)[x]/(x^n − 1)$. Consider the subset generated by these two fixed polynomials: $⟨f (x), g(x)⟩ = ...
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1answer
149 views

multiplying ideals

At the moment I am learning algebraic number theory and I have seemed to be missing some basic understanding. How do you multiply ideals For example $$(2,1+\sqrt{-5})^2 = ...
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1answer
71 views

Need an explanation for homomorphism in commutative algebra

I'm self-learning commutative algebra following "Introduction to Commutative Algebra". When dealing with concepts like "contraction" and "extension", some exercises in this book don't specify which ...
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65 views

What's the point of defining left ideals?

I admit, I haven't gotten really far in studying abstract algebra, but I was always curious (ever since I saw a definition of an ideal) why is the notion of left-sided ideal introduced when we ...
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1answer
52 views

Isomorphism between rings

Let $R$ be the ring of real valued continuous functions defined on the interval $[0, 1]$. Let $I = \left\lbrace f \in \mathbb{R} : f^2(0) + f^2(1) = 0 \right\rbrace$. 1) Prove that $I$ is an ideal. ...
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Prove $M$ is a Maximal Ideal in $\Bbb Z\times \Bbb Z$

A problem from introduction to abstract algebra by Hungerford. It asks: If $p$ is a prime integer, prove that $M$ is a maximal ideal in $\mathbb Z \times \mathbb Z$, where $M =\{(pa,b)\mid a,b\in ...
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239 views

Equivalence Of Definitions Of Prime Ideal In Ring Without $1$

Let $R$ be a rng, so that $1\not\in R$. I am trying to show that following are equivalence of definition of prime ideal $P$; i) $AB\subseteq P$ with $A,B\subseteq R$ implies $A\subseteq P$ or ...
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86 views

Is the ideal $I=(x_1 x_5 - x_2 x_4 , x_1 x_6 - x_3 x_4)$ of $k[x_1,…,x_6]$ a radical ideal? Is it a prime ideal?

Is the ideal $I=(x_1 x_5 - x_2 x_4 , x_1 x_6 - x_3 x_4)$ of $k[x_1,...,x_6]$ a radical ideal? Is it a prime ideal? thanks
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249 views

Finitely many prime ideals lying over $\mathfrak{p}$

Let $A$ be a commutative ring with identity and $B$ a finitely generated $A$-algebra that is integral over $A$. If $\mathfrak{p}$ is a prime ideal of $A$, there are only finitely many prime ideals ...
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1answer
70 views

Necessary and sufficient condition for $r(\mathfrak a)$ to be prime

As we know, $$\mathfrak a~\text{is a primary ideal}\Rightarrow r(\mathfrak a)~\text{is a prime ideal}. $$ But $r(\mathfrak a)$ may not be a prime ideal if $\mathfrak a$ isn't a primary ideal. ...
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142 views

Checking the maximality of an ideal

Let $R = \mathbb{Z}_{(2)}$ be the localization of $\mathbb{Z}$ at the prime ideal generated by $2$ in $\mathbb{Z}$. Then prove that the ideal generated by $(2x-1)$ is maximal in $R[x]$. Otherwise ...
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117 views

Find all elements of quotient ring

I am studying the definitions of rings, ideals, and quotient ring, but I have a bit problem to apply the theory into the practice. I would like to find all elements of quotient ring $\mathbb{Z}[i]/I ...
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Question about Principal Ideals

I'm just learning basic ring theory and had a question about the definition of a principal ideal. For a commutative ring $R$ with unity, Fraleigh defines the principal ideal generated by $a\in R$ as ...
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1answer
127 views

What are the maximal ideals of $\mathbb{Z}[t,t^{-1}]\otimes \mathbb{Q}$?

I know that $\mathbb{Z}[t,t^{-1}]$ is a localization of $\mathbb{Z}[t]$, the multiplicative set consisting of the non-negative powers of $t$. But I do not know the maximal ideals of ...
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1answer
31 views

Show $I=p\mathbb{Z}$ for prime $p$.

Let $I\subset\mathbb{Z}$ be an ideal such that $I\neq \mathbb{Z}$ and if $I\subset J\subset\mathbb{Z}$ then $I=J$ or $J=\mathbb{Z}$. Show that $I=p\mathbb{Z}$ for some prime $p$. Attempt: We know ...
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21 views

Explanation why $R_P=(S^{-1}R)_{P(S^{-1}R)}$

Suppose $R$ is a ring and $P$ a prime ideal. If $S$ is a mutliplicative subset, can anyone explain why we have the equality $R_P=(S^{-1}R)_{P(S^{-1}R)}$ when seen as subsets of the quotient field of ...
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44 views

Free modules and ideals

I am trying to show that an ideal I of R=$\mathbb{C}[x_1,x_2]$ generated by $x_1, x_2$ is free R-module. I am trying to show that I has a basis of the two generators given above. But I am not able to ...
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Number of maximal and prime ideals

Find how many prime and maximal ideals there are in the ring consisting of matrices $$M= \begin{bmatrix} a & b & c \\ 0 & a & b \\ 0 & 0 & a \\ \end{bmatrix} $$ ...
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329 views

Is quotient of a ring by a power of a maximal ideal local?

Say I have a commutative ring $R$ with a maximal ideal $m$. Then $m/m^k$ is a maximal ideal in $R/m^k$ for any $k$. Is it the only maximal ideal, i.e. is $R/m^k$ a local ring? This is a well ...
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1answer
34 views

Does $I(J\cap K)=IJ\cap IK$ hold in a Dedekind ring?

For ideals in any ring, we have the relation $I(J\cap K)\subseteq IJ\cap IK$. Do we actually have equality if we are in a Dedekind domain? I've been looking around for a reference, but haven't found ...
3
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1answer
83 views

Confused on a proof that $\langle X,1-Y\rangle$ is not principal

I'm getting stuck on a passage in my notes. The claim is that the ideal $P=\langle X,1-Y\rangle$ is not principal in $\mathbb{Q}[X,Y]/\langle 1-X^2-Y^2\rangle$. This follows since $P^2=\langle ...
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2answers
84 views

Quotient ring is cyclic group implies every ideal is generated by 2 elements

I'm trying to solve the following exercise: Let $R$ be a commutative ring with identity. If for every ideal $\mathfrak{a} \neq 0$ of $R$ we have ($R/\mathfrak{a}$,+) is a cyclic group then ...
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1answer
22 views

Showing the $C^*$ identity

I'm working through a proof in Dixmier's book on $C^*$-algebras and I'm stuck on part of a proof. I'm given a Banach algebra $\mathcal{A}$ which has norm $\lVert\cdot\rVert$ and a semi-norm ...
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643 views

Tensor product of quotient rings [duplicate]

$A$ is a commutative ring with unit and $\mathfrak a$, $\mathfrak b$ ideals. I have to show that $$A/\mathfrak a \otimes_{A} A/\mathfrak b \cong A/(\mathfrak{a+b}).$$ Any hint ?
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1answer
63 views

$A_{p}$ is a field when $p$ is a minimal prime and $A$ reduced

$A$ is a reduced commutative ring with unit; $p$ is a minimal prime ideal. If $S = A \setminus{p}$ , I have to show that the ring $A_{p} = S^{-1}A$ is a field. My thoughts: Since $p$ is a minimal ...
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36 views

Proof for uniqueness for ideal multiplication

I am across the following question here: The uniqueness of a special maximal ideal factorization Let R be a domain, and let I be an ideal that is a product of distinct maximal ideals in two ways, ...
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144 views

Proof for maximal ideals in $\mathbb{Z}[x]$

I have been trying to prove the following theorem: Every maximal ideal in $\mathbb{Z}[x]$ has the form $(p, f(x))$ where $p$ is prime integer and $f$ is primitive integer polynomial that is ...
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130 views

Calculate the primary decomposition

Consider the polynomial ring $R=K[x_1,\ldots, x_8]$ over field $K$. Set $\mathfrak{p}_1=(x_1, x_2, x_5, x_6)$, $\mathfrak{p}_2=(x_3, x_4, x_7, x_8)$ and $I=\mathfrak{p}_1\cap \mathfrak{p}_2$, ...
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1answer
65 views

Radical of an ideal - prove every prime ideal that contains $I$ also contains $\sqrt{I}$

Let $R$ be a commutative ring with a unit and $I$ an ideal. Please prove that every prime ideal that contains $I$ also contains $\sqrt{I}$. I easily conclud that $I \subseteq \sqrt I$ but I ...
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1answer
48 views

Minimal prime ideals consist of zerodivisors [duplicate]

I don't find the proof for this little demonstration ... Let $P$ be a minimal prime ideal of $A$. Show that $P$ is contained in the set of zero divisors of $A$.
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84 views

Ideal of a Vanishing set $I(V(F[X,Y]))$ and how to repeat the computation.

The video I am getting this from is found here: https://www.youtube.com/watch?v=spHxUPvrkXw, it is around 5 minutes in. The first part of the question is: for $F[X,Y] = Y^2 - X^3 = 0$ find ...
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1answer
106 views

Invertible elements and maximal ideals of a localization

Let $n\in\mathbb Z$ and let $A$ be the set of integers co-prime to $n$. Denote $A^{-1}\mathbb Z$ by $\mathbb Z_{(n)}$. 1) Find the invertible elements of $\mathbb Z_{(6)}$ My attempt: let $m$ be ...
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1answer
62 views

Question about comaximal ideal proof

Let $A$ be a ring and $M\subseteq A$ a maximal ideal. Show that if $I\subseteq A$ such that $I\not\subseteq M$, then $M$ and $I$ are comaximal($M+I=A$). I cannot find the proof for this statement.