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, ...

learn more… | top users | synonyms

0
votes
3answers
37 views

Let $R$ be a PID and $I$ is a non zero proper ideal of $R$. show that if $R/I$ has no nonzero zerodivisor, then it is a field. [on hold]

Let $R$ be a PID and let $I$ be a non-zero proper ideal of $R$. Show that if $R/I$ has no non-zero zerodivisor, then it is a field.
1
vote
1answer
55 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 ...
3
votes
2answers
47 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 ...
1
vote
1answer
33 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. ...
3
votes
2answers
61 views

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 ...
-2
votes
0answers
26 views

some questions about ideals [on hold]

I am educated in the field of mathematical analysis. I need some information about semigroups: What are all ideals of $(R^+ , +)$. {I think that $(0,a)$} and also of $[0,1]$ with multiplication ...
6
votes
2answers
71 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
1
vote
1answer
49 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. ...
0
votes
2answers
65 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 ...
0
votes
1answer
20 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 ...
1
vote
0answers
39 views

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 ...
3
votes
0answers
61 views

is there a criterion that says whether an ideal is radical or not?

Let $R=k[x,y,z]$. Is there a criterion that says whether an ideal of $R$ is radical or not? thanks
3
votes
1answer
112 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 ...
1
vote
1answer
25 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 ...
0
votes
0answers
15 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 ...
1
vote
1answer
26 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 ...
-1
votes
1answer
77 views

Number of maximal and prime ideals

Find how many prime and maximal ideals there are in the ring consisting of matrices $$M= \begin{bmatrix} a & b & c \\ 0 & a & b \\ 0 & 0 & a \\ \end{bmatrix} $$ ...
3
votes
2answers
82 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 ...
1
vote
1answer
18 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
votes
1answer
70 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 ...
1
vote
2answers
41 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 ...
1
vote
1answer
14 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 ...
0
votes
2answers
75 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 ?
2
votes
1answer
45 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 ...
0
votes
0answers
23 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, ...
1
vote
0answers
30 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 irreducible ...
1
vote
0answers
73 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 $I=\mathfrak{p}_1\cap \mathfrak{p}_2$, ...
0
votes
1answer
27 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 ...
-1
votes
1answer
66 views

Maximal multiplicative set and minimal prime ideal

Let $A$ be a ring and $P$ a prime ideal included in $A$. Show that $A \setminus P$ is a maximal multiplicative set if and only if $P$ is a minimal prime ideal of $A$. What can be the proof for this ...
0
votes
1answer
27 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 $A$.
2
votes
0answers
26 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 ...
0
votes
1answer
56 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 ...
0
votes
1answer
33 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.
0
votes
1answer
24 views

Show that these rings of Gaussian integers are ideals in $\mathbb{Z}[i]$?

Consider the ring of Gaussian integers: $\mathbb{Z}[i]$ = {a + bi | a, b ∈ Z} ⊂ $\mathbb{Q}[i]$ with $i^2$ = −1. Let I = $(2+3i)$ and J =$(2−3i)$. Show that I and J are ideals of $\mathbb{Z}[i]$.
2
votes
0answers
48 views

Example of irreducible ideal which is not strongly irreducible

I have read a paper with title Ideal Theory in Commutative Semirings by Reza Ebrahimi Atani and Shahabaddin Ebrahimi Atani. In this paper we have the following definitions: An ideal I is irreducible ...
2
votes
1answer
53 views

Maximal ideals in $\mathbb{Z}[x]$

I am trying to solve the following problem from Artin: Every maximal ideal $\mathbb{Z}[x]$is of the form $(p,f)$ where p is a prime integer and $f$ is a primitive polynomial that is irreducible modulo ...
1
vote
0answers
27 views

Class Group of Ring of Integers of $\mathbb{Q}[\sqrt{-57}]$

Let $R$ denote the ring of integers of the imaginary quadratic number field $\mathbb{Q}[\sqrt{-57}]$. I must find the ideal class group $\mathcal{C}$. Using the Minkowski Bound, I know that I need ...
2
votes
0answers
49 views

Normal ring and unmixed ideals

Let $R$ be a commutative Gorenstein local ring , $I$ an ideal of $R$ . If $R/I$ is normal ring , then for any $p \in \operatorname{Ass_{R}}(R/I)$, $\operatorname{ht}(p)= \operatorname{ht}(I)$?
4
votes
2answers
69 views

If $A/\mathfrak a$ is flat over $A$ then $V(\mathfrak a)$ is open. Why?

I am trying to understand the following statement. Let $A$ be a noetherian commutative ring and $\mathfrak a\subset A$ is an ideal. Suppose that the ring $A/\mathfrak a$ is flat over $A$, then ...
0
votes
1answer
37 views

Ideals in direct product of rings

I am trying to solve that problem: Let $ R_1,...,R_n$ rings with identity, every ideal of $R=\prod_{i=1}^n R_i$ is in form $\prod_{i=1}^n I_i$ where $ I_i$ideal of $R_i$. The first part is clearly if ...
2
votes
1answer
63 views

Minimal primes and zero divisors

Let $R$ be a commutative local ring, $M$ a finitely generated $R$-module, and $x \in M$. Is it true that if for any $p \in$ $\operatorname{Min}(R)$ there exists $a_{p}\notin{p}$ such that $a_{p}x=0$, ...
0
votes
2answers
34 views

Spec($A$) is connected if $A$ is local

Another exercise from Balwant-Singh: Show that if $A$ is local then Spec($A$) is connected in the Zariski topology. Any hint ?
1
vote
1answer
42 views

Idempotent/Spec

I'm studying Basic Commutative Algebra by Balwant-Singh; I'm stuck on this exercise: $A$ is a commutative ring; show this $3$ conditions are equivalent: 1) $A$ contains a non-trivial idempotent 2) ...
0
votes
1answer
53 views

Is it true that an ideal is primary iff its radical is prime?

Is it true that an ideal $I$ in a commutative ring is primary iff $Rad(I)$ is prime? If not, what are some nice counterexamples?
1
vote
1answer
32 views

When an Intersection of Prime Ideals is a Prime Ideal

Let $R$ be an arbitrary ring, $\{P_1,....,P_n\}$ be a set of prime ideals. Verify that $P_1 \cap ... \cap P_n$ is prime if and only if there exists $1 \leq i \leq n$ such that $P_i$ is contained in ...
7
votes
2answers
95 views

Techniques for showing an ideal in $k[x_1,\ldots,x_n]$ is prime

An affine variety $X$ over a field $k$ is irreducible if and only if its defining ideal $I(X)$ is prime (in this post we use the convention that varieties are not necessarily irreducible). Hence, it ...
3
votes
1answer
39 views

Complexifications of degree 3 subschemes in $\mathbb A^2_{\mathbb R}$

I am trying unsuccessfully to solve exercise II-20 (page 65) from the book "The geometry of schemes" by Eisenbud and Harris. In this exercise it is stated that there are two non-isomorphic subschemes ...
1
vote
2answers
42 views

Prime ideal in a ring with some property is maximal

Let $R$ be a ring where for every element $a \in R$, there exists a positve integer $n_a \gt2$ such that $a^{n_a}=a$. Prove that every prime ideal in $R$ is maximal. I think that I would want to ...
1
vote
1answer
70 views

When does coprimality carry over to the base ring in an extension of Dedekind domains?

Let $A$ be a Dedekind domain. Let $K$ be the field of fractions of $A$ and $L$ is some finite field extension of $K$. Then let $B$ be the integral closure of $A$ in $L$. (Sorry I don't know how to ...
1
vote
0answers
24 views

Question on radical ideals

i need help with showing that I = <xy, xz, yz> is a radical ideal. Thanks