0
votes
1answer
24 views

The nil-radical is an intersection of all prime ideals proof

Every proof I found online made the same implications. Take one for example: http://www.artofproblemsolving.com/Wiki/index.php/Nilradical I'm quoting the relevant part, which confuses me: "To show ...
4
votes
0answers
69 views

Counterexamples for lcm-gcd identity and modular law for rings

In Mile Reid's Undergraduate Commutative Algebra, Exercise 1.3, we need to find counterexamples of lcm-gcd identity and modular law in the ring $A=k[X,Y]/(XY)$: $(I+J)(I\cap J)=IJ$; ...
1
vote
1answer
88 views

Quotient ring isomorphism

I think that if $A$ is any commutative ring with unity and $q\in A$, $p\in A[x]$ then we have $A[x]/(q,p)\cong A/(q)[x]/(\bar{p})$ where $\bar{p}$ denotes the class of $p$ in $A/(q)[x]$. Is this ...
0
votes
1answer
17 views

Finding a particular principal open subset of $Spec R$

Let $V\subseteq U$ be open subsets of $X=\text{Spec } R$, where $R$ is a commutative ring. So $V$ is the set of prime ideals not containing some ideal $I$, and $U$ is the set of prime ideals not ...
-2
votes
0answers
22 views

locally unital ideals of ringa [duplicate]

1 down vote favorite Let R be a ring with unity not necessarily commutative and I an ideal of R.Let for every element a of I there exists an element c of I such that ac=a.Note that c is related to a. ...
3
votes
1answer
56 views

Locally unital ideals [duplicate]

Let $R$ be a ring with unity not necessarily commutative and $I$ an ideal of $R$. Let for every element $a \in I$ there exists an element $c\in I$ such that $ac=a$. Note that $c$ is related to $a$. ...
0
votes
0answers
38 views

Prime ideal is contraction of prime ideal iff it's saturated

Let $\varphi: A\to B$ be a commutative ring homomorphism and $P$ a prime ideal of $A$. The expansion of an ideal $I\subset A$ is the ideal generated by $\varphi(I)$ in $B$, and the contraction of an ...
2
votes
1answer
34 views

Which is a subring? Which is an ideal?

We are having ring $\mathbb{Z}[\sqrt{-6}]$. Which of the sets is subrings of $\mathbb{Z}[\sqrt{-6}]$ and which are ideals? $\mathbb{Z}+5\mathbb{Z}[\sqrt{-6}]$ $5\mathbb{Z}+\mathbb{Z}[\sqrt{-6}]$ ...
0
votes
2answers
50 views

Maximal ideal which isn't principal

Let $J=(x-2,x-y^2-3)$ ideal in the polynomial ring $\Bbb R[x,y]$. Please help me prove that $J$ is a maximal ideal which isn't principal, and that $\Bbb R[x,y]/J \cong \Bbb C$.
1
vote
0answers
40 views

Ideals of $\Bbb Z/p^2q\Bbb Z$

Let $p,q$ be distinct primes. Then $\mathbb{Z}/p^2q\mathbb{Z}$ has 3 distinct ideals. $\mathbb{Z}/p^2q\mathbb{Z}$ has 3 distinct prime ideals. $\mathbb{Z}/p^2q\mathbb{Z}$ has 2 distinct prime ...
0
votes
1answer
24 views

Specific Ideal determinations for a Ring

Uploaded in a picture, rather than typing it all. Note that these are unmarked questions from a sample exam. Just trying to study, but have forgotten almost everything (three major exams before this ...
1
vote
1answer
15 views

Determining if any of these three are an ideal of $\mathbb{R}[x]$

$\mathbb{R}[x]$ denotes the ring of polynomials in $x$ with real coefficients. Let $I \subset \mathbb{R}[x]$ be the subset of those polynomials with constant coefficient $0$, and let $J \subset ...
1
vote
1answer
34 views

Proving $M_p$ is maximal in $C[0,1]$

Let $M_p$ be the ideal of those continuous functions of $C[0,1]$ which have $p\in [0,1]$ as a zero. It is a commonly known fact that $M_p$ is a maximal ideal. However, the proof is generally ...
1
vote
1answer
109 views

Subset of $\mathbb{Z} \times \mathbb{Z}$

I have a past exam question that is as follows: Let $k$ be a fixed integer and $S = \{(a,ka)|a \in \mathbb{Z}\}$ be a subset of $\mathbb{Z} \times \mathbb{Z}$. Prove that $S$ is a subgroup of ...
3
votes
0answers
24 views

Need help with finding generator

$I=\{a+bi \in R\mid a \equiv b\pmod 2\}$ is ideal of $R=Z[i]=\{a+bi\mid a,b \in Z\}$. Can somebody help me to find the generator of $I$?
0
votes
1answer
58 views

If $R/I$ satisfies Serre's condition $S_2$ then $R/(I,x)$ satisfies $S_1$

Let $I$ be an ideal of polynomial ring $R=K[x_1,\ldots,x_n]$ and $x$ be a non-zero divisor of $R/I$. Is the following statement true? If $R/I$ satisfies Serre's condition $S_2$ then $R/(I,x)$ ...
0
votes
4answers
73 views

Maximal ideal in the ring of polynomials over $\mathbb Z$

Let $\mathbb Z[x]$ the ring of polynomials with integers coefficients in one variable and $I =\langle 5,x^2 + 2\rangle$, how can I prove that $I$ is maximal ideal. I tried first see that $5$ and ...
1
vote
1answer
42 views

Ideals problem from Hungerford Algebra

This question is from Hungerford's Algebra. Let $R$ be a ring with identity and $S$ the ring of all $n\times n$ matrices over $R$. $J$ is an ideal of $S$ iff $J$ is the ring of all $n\times n$ ...
5
votes
3answers
121 views

Commutative ring with an ideal that contains all the nonunits

Is there an example of a commutative ring with an ideal that contains all the non-units? I was trying to think of some subring of $\mathbb Q$, but I couldn't get it to work.
0
votes
1answer
34 views

A local subring of $F[[x]]$?

Suppose that $F$ is a field and $R=F⊕x^2F[[x]]$, where $F[[x]]$ is the ring of power series in one indeterminate $x$ with coefficients in $F$. I guess that $R$ is a local ring with the maximal ...
1
vote
1answer
26 views

Prove that if $\mathrm D$ is integral domain, then $\mathrm D$is UFD iff these conditions hold.

Prove that id $\mathrm D$ is integral domain, the "$\mathrm D$ is UFD " iff (1) $\mathrm D$ satisfies the ACC for principal ideals. (2) every irreducible element is a prime element. ...
0
votes
0answers
22 views

Complexity of and an algorithm for finding ideals of a ring?

One of the problems that has been a roadblock in my understanding of ideals has been how one would find them. One way of finding an I of some ring R would be to say $ \forall x \in I, \forall r \in R ...
2
votes
1answer
34 views

ideal calculations: $2\mathcal{O}_K=\mathfrak{B}^4$ in the ring of integers of $K=\mathbb{Q}(i,\sqrt{2m})$

Let $K=\mathbb{Q}(i,\sqrt{2m})$ where $m \in \mathbb{Z}$ is odd and squarefree. Let $\alpha = (1+i)\sqrt{2m}/2$. Then $\alpha^2=im$, such that $\alpha$ is part of the ring of integers $\mathcal{O}_K$. ...
0
votes
0answers
61 views

An $n$-generated ideal of grade $n$ can be generated by an $R$-sequence in any order

It is known to me that if $I$ is an $n$-generated ideal of a commutative ring $R$ with $\operatorname{grade}(I)=n$, then it is generated by an (ordered) $R$-sequence in $I$ of length $n$. I have a ...
4
votes
1answer
48 views

Do $IJ$ and $I\cap J$ coincide if $I$ and $J$ are coprime? Also if ring $R$ has a $1$ and is not commutative?

Let $R$ be a ring (with identity) and let $I,J$ be two coprime (two-sided) ideals in it. In Algebra: Chapter $0$, Aluffi, III. exercise 4.5. the reader is asked to prove that: $$IJ=I\cap J$$ ...
0
votes
2answers
31 views

Let $R$ be a commutative ring with identity. Let $M$ be an ideal such that every element of $R-M$ is a unit. Then $R/M$ is a field.

Let $R$ be a commutative ring with identity. Let $M$ be an ideal such that every element of $R$ not in $M$ is a unit. Then $R/M$ is a field. I am solving this question of NBHM 2011. To solve this is ...
2
votes
1answer
33 views

Isomophism between rings an two right ideals

Let I, J two right ideals of a ring R such that I+J =R. Show thath the direct sum of I and J is isomorphic to the direct sum of R and the intersection of I and J. Can anyone please give me at least ...
0
votes
2answers
36 views

Not so usual equivalence of maximal left ideal of a ring

I was reading Foundations of Module and Ring Theory and i found this equivalence of maximal left ideal as exercise in the the first chapter: A left ideal $I$ of a ring $R$ is a maximal if and only ...
1
vote
3answers
57 views

$a^n = 0 \implies a \in P$ (where $P$ is a prime ideal)

Is the above true? (I think it is!) if so, please can somebody explain why? I don't see it!
3
votes
0answers
37 views

When are all (prime) ideals of an $R$-algebra, extensions of (prime) ideals of $R$?

Let $f:R\rightarrow R'$ be a homomorphism of commutative noetherian rings. When are all (prime) ideals of $R'$ extensions of (prime) ideals of $R$? Is it true for the case $R'$ is $R$-flat?
0
votes
1answer
28 views

principal ideals, integral domains, ideals,?

I am stuck trying to grasp this concept. I know that $\Bbb{Z}$ is a PID, $R=\Bbb{Z}[X]$ is not a PID, $\Bbb{Z}[i]$ is a PID. If someone could help me grasp these concepts it would be helpful. ...
0
votes
2answers
22 views

Finding a ring isomorphism

Let $\phi : R \to R'$ be a ring epimorphism and $J\lhd R'$ an ideal of $R'$. Indicate a ring isomorphism $\psi: R/\phi^{-1}(J) \to R'/J$ The only thing i know about this problem is that ...
0
votes
1answer
14 views

About the connection between ideals and homomorphisms

I know that for a homomorphism of rings $\psi : R\rightarrow S$ we have that $\ker\psi$ is an ideal of $R$. I was wondering if the opposite direction is true: Let $I$ be an ideal of $R$. Then does ...
2
votes
1answer
40 views

Ideals generated for commutative ring

The following is a problem from the Gallian book. I'm trying to understand what exactly this ideal is and how to verify that it is in fact an ideal. "Let $R$ be a commutative ring with unity and let ...
0
votes
0answers
59 views

What are the open and closed sets in $\operatorname{Spec}\mathbb{C}[x,y]/(y^2-x^3)$?

What are the open and closed sets in $\operatorname{Spec}\mathbb{C}[x,y]/(y^2-x^3)$ ? $\operatorname{Spec}\mathbb{C}[x,y]/(y^2-x^3)=\{ (0),\ (\tilde{x}-a,\tilde{y}-b),\ b^2=a^3\}$.
1
vote
1answer
47 views

Symmetric powers of ideal quotients in a local ring.

Let $R$ be a local ring and $I \subset R$ any ideal. When is it the case that $(I \: \backslash I^2)^n = I^n \: \backslash I^{n+1}$? Put another way, when is the natural map $\text{Sym}^n(I/I^2) ...
2
votes
1answer
33 views

Principal ideals as generated groups

This seems like a pretty simplistic question, but I can't find a solid, non-ambiguous answer to it. The question I'm given: Is $I$ a principal ideal of $R$? Given: $R=\mathbb{Z}$ and $I=\left\langle ...
0
votes
1answer
49 views

Addition and Multiplication table for Ring/Ideal

I'm not sure if it's possible to show it here, but how would the addition and multiplication table look like for R/I (where R is rings with ideal I) when $$ R = Z_{12} \text{ and } I = \{0,3,6,9\} ...
4
votes
2answers
72 views

Computing the radical of an ideal

What is the best way to compute $\sqrt{(X^2-YZ,X(1-Z))}$ ? This is after using Nullstellensatz by the way as I thought it would be easier to compute a radical than finding the vanishing ideal.
1
vote
0answers
27 views

A problem involving ideals and prime ideals. [duplicate]

Please help me with a solution to this problem. Let $R$ be a commutative ring. Let $A_1, A_2$ be two ideals of $R$, and $P_1, P_2$ two prime ideals of $R$. Assume that $A_1 \cap A_2 \subseteq P_1 ...
1
vote
2answers
68 views

When is nilradical not a prime ideal

Atiyah gives this criterion for nilradical to be a prime ideal.Nilradical is the intersection of prime ideals.Is nilradical prime iff there is only one prime ideal? ie Intersection of distinct prime ...
2
votes
2answers
90 views

Number of elements in the ring $\mathbb Z [\iota]/\langle 2+2 \iota\rangle$

The question is : Show that $I=\langle 2+2 i\rangle$ is not a prime ideal of $\mathbb Z[i]$. Also find the number of elements in $\mathbb Z[i]/I$ and its characteristic. My try: I started with ...
1
vote
0answers
41 views

Ideals - A Geometric Interpretation?

The standard way to define an ideal is as follows: $I$ is an ideal if it satisfies the following conditions: $(I,+)$ is a subgroup of $(R,+)$ $\forall x \in I$, $\forall r \in R :\quad ...
1
vote
1answer
44 views

$(2,1+\sqrt{-5})$ has integral basis $2$, $1+\sqrt{-5}$

$2,1+\sqrt{-5}$ is an integral basis for the ideal generated by them in $\mathbb{Z}[\sqrt{-5}]$. Is there a quick way to see this? What if these two are replaced with another pair? My method: Write ...
0
votes
1answer
62 views

Equality with powers of an ideal

Let $A$ be an arbitrary (commutative with an identity) ring. Suppose $\alpha$ is an ideal. Is it true that $$\alpha(\alpha\cap\alpha^2\cap\alpha^3\cap…)=\alpha\cap\alpha^2\cap\alpha^3\cap…?$$ ...
2
votes
2answers
53 views

Example of Localization and Prime Ideals

For each $n\in \mathbb Z^+$, give an example of a localization of $\mathbb Z$ with exactly $n$ prime ideals. Justify your answer. Could an example have something to do with a UFD or Noetherian ...
0
votes
2answers
65 views

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 ...
1
vote
2answers
22 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 ...
1
vote
1answer
49 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 ...
1
vote
1answer
113 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 ...