0
votes
1answer
12 views

Question about the wording of a Ring Theory problem involving ideals

The homework question is: If $I,J$ are ideals of $R$, let $IJ$ be the set of all sums of elements of the form $ij$, where $i \in I$, $j \in J$. Prove that $IJ$ is an ideal of $R$. The phrase "the ...
1
vote
0answers
17 views

How to prove a subset is an ideal

I just started learning about Ring Theory today and I am having some trouble truly understanding and being able to apply certain concepts. The first concept I am having trouble understanding is an ...
2
votes
1answer
30 views

The ideal $\langle x,y \rangle$ in $F[x,y]$ is not principal.

Let $F$ be a field. Apparently we know that $\langle x,y \rangle \neq \langle g(x,y) \rangle$ for any $g \in F[x,y]$. Why is this the case?
0
votes
1answer
31 views

Operations with ideals in a commutative ring

Let $R$ be a commutative ring with identity. Let $A$ and $B$ be ideals in the ring $R$. It is true that $(A\cap B)(A+B)$ equals the product $AB$?
0
votes
4answers
103 views

Is product of prime ideals prime?

I'm trying to show that the product of ideals $(x_1, x_3)$ and $(x_2, x_4)$ in $\mathbb C[x_1, x_2, x_3, x_4]$ is a radical ideal, but no other way that I can think of works. So, is the product ...
3
votes
1answer
112 views

Questions related to maximal ideals

In my previous sessional exams, I was asked to prove these two: 1) Find a ring which doesn't have a maximal Ideal. 2) If a ring has unity, then it has a maximal Ideal. About the ...
0
votes
1answer
32 views

Showing that an ideal is prime

I think that $k[x,y,z]/(z-1, x^2-y)$ can be identified as a subset of $k[x,y]$ with all polynomials whose $x$ terms are only degree one. Therefore I conclude that $k[x,y,z]/(z-1, x^2-y)$ is integral ...
1
vote
1answer
30 views

no. of elements in $\mathbb Z[i]/\langle 3+i\rangle$. [duplicate]

$\mathbb Z[i]/\langle 3+i\rangle$ can be represented as :$\{a+3b+\langle 3+i\rangle\big|~~a,b\in \mathbb Z\}$ How shall I find the total no. of elements in $\mathbb Z[i]/\langle 3+i\rangle$.. ...
1
vote
1answer
21 views

$A=\{(3x,y)\mid x,y\in \mathbb Z\}$ is a maximal ideal of $\mathbb Z \oplus \mathbb Z$

I've a question from gallian which states: Show that $A=\{(3x,y)\mid x,y\in \mathbb Z\}$ is a maximal ideal of $\mathbb Z \oplus \mathbb Z$.Generalize.What happens if $3x$ is replaced by $4x$... ...
0
votes
1answer
33 views

Doubt regarding zero elements in factor ring :$\mathbb Z[i]/\langle3-i\rangle$

I have the factor ring $\mathbb Z[i]/\langle3-i\rangle$ and am asked to find elements zero in this ,they are $0,3-i,i(3-i),(3-i)+i(3-i)$. But I can't understand how do we guarantee these are the only ...
0
votes
1answer
22 views

An Ideal which is Maximal additive subgroup is a Maximal Ideal

How should I prove this: Any Ideal which is a Maximal additive subgroup is also a Maximal Ideal . any idea how to prove it..
1
vote
2answers
27 views

An ideal which is not finitely generated

Let $K$ be a field and $A=K[x_1,x_2,x_3,...]$. Prove that the ideal $I:=\langle x_i: i \in \mathbb N\rangle$ is not finitely generated as $A$-module. I have no idea what can I do here, I mean, ...
0
votes
1answer
32 views

A question regarding the number of generators of an ideal [duplicate]

Let $I$ be an ideal in $\mathbb{C}[x_1 ,x_2 ,x_3 ,x_4 ]$ such that $I$ is generated by $x_1 x_3$, $x_2 x_3$, $x_1 x_4$, and $x_2 x_4$. How to show that this I cannot be generated by two ...
0
votes
2answers
40 views

Showing that an ideal is maximal

Let $k$ be an algebraically closed field and $f$ be the polynomial $x_1x_2+x_2x_3+x_3x_1$ in $k[x_1, x_2, x_3]$. Here $f$ is irreducible. Then this polynomial ring is not a $PID$, it is only an ...
1
vote
0answers
34 views

Ideal and its number of generators [duplicate]

Consider an ideal $I$ in $\mathbb{C}[x_1, x_2, x_3, x_4]$ such that $I$ is generated by $x_1x_3, x_2x_3, x_1x_4,$ and $x_2x_4$. I think this ideal cannot be generated by two elements, but can't ...
1
vote
2answers
48 views

Sum and product of comaximal ideals

Let $R$ be a commutative ring with unity. If $R=I_{i}+I_{j}$, for all $i\ne j$, where $I_1,I_2,...,I_n$ are ideals of $R$, I want to show that $$R=I_{n}+I_{1}I_{2}\cdots I_{n-1}.$$ I started off ...
0
votes
0answers
26 views

Ideals of a field

I had the following - apparently straightforward - question on one of my past assignments: Show that a field has no other ideals except $\{0\}$ and the field itself. This was the proof I gave: ...
2
votes
2answers
134 views

How to show that there is a bijective correspondence between two sets of prime ideals

I'm trying to solve this Algebra Problem, and I'm not quite sure, if I'm on the right way. Let $R$ be a commutative ring and $S \subset R$ a multiplicative subset. Show that $p \to pS^{-1}R$ ...
1
vote
1answer
31 views

Localizing at maximal ideals and the product

Let $D$ be an integral domain, $M_{i}$, $i = 1,...,r$ be some of its mutually distinct maximal ideals, and $e_{i}$be positive integers for all $i$. Is it true in general that the extension of the ...
0
votes
0answers
23 views

the sum of two $z^0$-ideals even in $C(X)$ need not to be a $z^0$-ideal

I need to get an example of two $z^0$-ideals while their summation is not? What i know that the sum is a $z^0$-ideal or all of $C(X)$ if and only if $X$ is quasi F-space So i'm searching for an ...
1
vote
1answer
51 views

Factorization in noetherian domains

I changed the title (and the body) of this question page, since user26857 provided a nice answer for my original question in a more general setting. Here's what the accepted answer below provides: ...
0
votes
0answers
73 views

Irreducible ideals that are not primary.

In my advanced algebra course I've heard that in a noetherian (commutative) ring every irreducible ideal is primary. Can you give a counter example in a non noetherian ring? I've been lookin' ...
1
vote
1answer
44 views

Does $ax\in\mathfrak{m}I$ with $x\in I\setminus\mathfrak{m}I$ and $a \in R$ imply $a\in\mathfrak{m}$ for an invertible fractional $R$-ideal $I$?

Let $R$ be an integral domain, $\mathfrak{m}$ a maximal ideal of $R$, and $I$ an invertible fractional $R$-ideal. If $x \in I \setminus \mathfrak{m}I$ and $a \not\in \mathfrak{m}$, do we have $ax ...
3
votes
1answer
42 views

Maximal Ideals in $R=\{a+bi:a,b\in \mathbb Z\}$

I've read similar question but please this is not duplicate of Maximal ideals in the ring of Gaussian integers because the answer to it contain PID which I've not yet done etc. $R=\{a+bi:a,b\in ...
0
votes
0answers
40 views

Meaning of $S^{-1}R$ notation

Here are objects defined in an exercise: Let $R$ be a commutative ring. Let $A$ be an ideal of $R$ and $S=\{1+a\mid a\in A\}$. The exercise then makes reference to the prime ideals of $S^{-1}R$. ...
1
vote
1answer
26 views

Example of ring with two maximal ideals such that the char of the quotients is $0$, respectively $p$.

I am looking either for an example of a commutative ring with identity and two maximal ideals, such that the characteristic of one of the quotient rings is finite and the other characteristic is zero, ...
1
vote
2answers
77 views

Prime ideals in $k[x,y]/(xy-1)$.

Let $k$ a field. Let $f$ be the ring injective homomorphism $$ f:k[x] \rightarrow k[x,y]/(xy-1)$$ obtained as the composition of the inclusion $k[x] \subset k[x,y]$ and the natural projection map $ ...
0
votes
1answer
24 views

A monomorphism from a ring to a direct sum

Let $R$ be a ring with a family of ideals $A_i$'s ($i\in I)$. We could consider a well-defined $R$-monomorphism from $R/∩A_i$ to the direct product of $R/A_i$'s sending $r+∩A_i$ to the tuple ...
1
vote
1answer
73 views

Finitely generated prime ideal and annihilator

Suppose $R$ is a commutative ring, $P$ is a prime ideal of $R$, $P$ is finitely generated, and $\operatorname{Ann}(P)=0$. Show that $$\operatorname{Ann}(P/P^2)=P.$$ These are my efforts: ...
1
vote
1answer
46 views

A relation in a finitely generated module

Suppose $R$ is a commutative ring, $I$ is an ideal of $R$, and $M$ a finitely generated $R$-module s.t. $M=IM$. How to prove: $$\exists a \in I \text{ such that } (1-a)M=0. $$ I tried to solve: ...
1
vote
3answers
63 views

Find the number of prime ideals (CSIR 2014)

Let $p,q$ be distinct primes. Then (1) $\dfrac{\mathbb{Z}}{p^2q}$ has exactly 3 distinct ideals. (2) $\dfrac{\mathbb{Z}}{p^2q}$ has exactly 3 distinct prime ideals. (3) $\dfrac{\mathbb{Z}}{p^2q}$ ...
0
votes
2answers
35 views

How to find the number of maximal ideals? [closed]

Let $n \geq 2$ and $n={p_1}^{e_1}{p_2}^{e_2}\cdots {p_r}^{e_r}$. Then the number of maximal ideal of $Z/nZ=Z_n$ is r n $e_1$+$e_2$+....+$e_r$ $p_1$$p_2$....$p_r$
8
votes
2answers
148 views

Determining consistency of a general overdetermined linear system

For $m > 2$, consider the $m \times 2$ (overdetermined) linear system $$A \mathbf{x} = \mathbf{b}$$ with (general) coefficients in a field $\mathbb{F}$; in components we write the system as ...
2
votes
2answers
93 views

Showing an ideal with maximality condition is prime.

Let $R$ be a commutative domain and suppose that $I \subseteq R$ is an ideal of $R$ maximal with respect to the property that $I^{-1} \not\subseteq R$. Show that $I$ is a prime ideal. This is ...
1
vote
2answers
24 views

Finding the element of the quotient ring $Z[i]/\langle 2+2i\rangle$

First, I'm writing an element to confirm whether I understood this quotient ring correctly. $$(5 + 7i) + \langle 2+2i \rangle = 2(2+2i) + (1+3i) + \langle 2+2i \rangle = ...
1
vote
1answer
47 views

For any two Ideals $A$ and $B$,$A+B=\langle A \cup B \rangle$

Below is the proof of : Prove that for any two ideals $A$ and $B$ of ring $R$,$A+B=\langle A \cup B~\rangle$ . Proof: By theorem (for any two ideals of a ring $R$ ,then the set $A+B$ is an ...
1
vote
1answer
43 views

Colon ideal of fractional ideals is itself a fractional ideal

I received this question on homework in my homological algebra class and I need some guidance. Let $R$ be a commutative integral domain and $K$ be its field of fractions. A fractional ideal $I$ of ...
5
votes
1answer
71 views

How to show that $\mathbb{C}[x_1,x_2,x_3, x_4]/(x_1x_2 - x_4x_3, x_1x_3 - x_2x_4, x_4x_1 - x_3x_2)$ is integral domain

I am looking for a way to show that the ring $\mathbb{C}[x_1,x_2,x_3, x_4]/I$ where $I = (x_1x_2 - x_4x_3, x_1x_3 - x_2x_4, x_4x_1 - x_3x_2)$ is an integral domain. In other words I want to show $I$ ...
2
votes
2answers
40 views

Prove that, if $(a)=(a')$, then $a'=ua$

Let $R$ be integral domain. Show that if $2$ principal ideals $(a)$ and $(a')$ are equal (where $a,a'\in R$) then there exists $u\in R^{\times}$ such that, $a'=ua$ Now if $(a)=(a')$ then ...
1
vote
2answers
58 views

Commutative ring and maximal ideal problem

Let $A$ be a commutative ring and $M$ be a proper maximal ideal in $A$. Show the following properties: (a) If each $a \in A \setminus M$ is a unit element in $A$, then $M$ is the only maximal ideal ...
1
vote
1answer
69 views

Intersection of two polynomial ideals

In the $4$-dimensional affine space $\mathbb{A}^4$ with coordinates $x,y,z,t$, consider $X$ as the union of the planes $$ X'=\{x=y=0\} $$ and $$ X''=\{z=x-t=0\} $$ (I'm working on a algebraically ...
-1
votes
2answers
116 views

Show that every maximal ideal in $ \mathbb{Z}[x, y] $ contains a prime number [closed]

Let $\mathfrak{M} \subseteq \mathbb{Z}[x, y]$ be a maximal ideal. Show that $ \exists\ p \in \mathbb{Z}$, $p$ prime such $p \in \mathfrak{M}.$ Thanks for the answers. I'd be interested in a proof ...
1
vote
1answer
54 views

Find all the ideals of $\mathbb Q[X]$

I am trying to find all the ideals of the ring $\mathbb Q[X]$. If $I$ is a non trivial ideal of $\mathbb Q[X]$, then there exists $p(x) \in \mathbb Q[X]$. Since $I$ is an ideal and a group under ...
1
vote
1answer
36 views

Left ideals of $M_n(K)$ [duplicate]

Let $K$ be a field and $n \in \mathbb N$. Show the following (i) Let $V \subset K^n$ be a subspace and $I_V$ the subset of $M_n(K)$ consisting of all the matrices whose rows belong to $V$. Prove that ...
0
votes
1answer
62 views

Problem on the number of generators of some ideals in $k[x,y,z]$ [on hold]

I have got stuck with two generator problems: The ideal $(zx,xy,yz)$ can't be generated by $2$ elements. The ideal $(xz-y^2,yz-x^3,z^2-xy)$ can't be generated by $2$ elements. Here the ...
2
votes
4answers
172 views

A finitely dimensional algebra over a field has only finitely many prime ideals all of them are maximal

Let $K$ be a field and let $R$ be a $K$-algebra with unity which is finite dimensional as a $K$-vector space. Prove that $R$ has only finitely many prime ideals all of which are maximal. (Hint: ...
1
vote
1answer
40 views

Product of two non-principal ideals

I have problems understanding why $$(6,2+\sqrt{-56})(6,-2+\sqrt{-56})=6(2,\sqrt{-56})$$ in $\mathbb{Z}[\sqrt{-14}]$. By definition the product of two ideals $$IJ=\sum_{i,j}^{k}f_{i}g_{j}$$ ...
2
votes
1answer
92 views

Analogy of ideals with Normal subgroups in groups.

I've started with Ideals in ring theory but still not comfortable with the analogy it has with normal subgroups in group theory.Like we can visualize normal subgroups as Is there some good intutive ...
0
votes
0answers
18 views

Size of a subset of the set of units of a quotient ring

Let $R$ be a commutative Dedekind domain with multiplicative identity $1$, let $k$ be a positive integer, and let $I$ be a nonzero proper prime ideal of $R$. Is there a way to find the size of the set ...
1
vote
2answers
65 views

Find a generator for an ideal in $\mathbb{Q}[T]$

Let $I$ be the ideal in $\mathbb{Q}[T]$ generated by $L=\{T^{2}-1, T^3-T^2+T-1,T^4-T^3+T-1\}$. Find $f\in\mathbb{Q}[T]$ such as $(f)=f\mathbb{Q}[T]=I$. The book solution proves that $I\subseteq ...