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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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: ...
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Rings and modules

Let $R$ be a ring in which every maximal ideal is a direct sum of cyclic $R$-modules. Now let $I$ be a proper ideal of $R$. What is the structure of $I$. Is it true that $I$ is a direct sum of cyclic ...
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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) $\dfrac{\mathbb{Z}}{p^2q}$ ...
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Which of the following is also an ideal?

If $U,V$ are ideals of a ring $R$, then which of the following is also an ideal of $R$? $U+V=\{u+v\mid u\in U,v\in V\}$ $U\cdot V=\{u\cdot v\mid u\in U,v\in V\}$ $U\cap V$ My attempt: I have ...
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A question from the Herstein's “noncommutative ring”, page 15, theorem 1.2.5

Let $R$ be a ring. $\rho$ is a maximal regular right ideal of $R$. $(R/\rho,+)$ is an irreducible $R$ module. $(r+\rho)\cdot r_1$ is defined by $rr_1+\rho$. $A$ is a two-sided ideal of $R$. ...
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If $p \in \operatorname{Ass}M$, then $R/P \subset M$.

Let $R$ be a commutative ring with unity. $M$ an $R$-module. Then $P \in \operatorname{Ass}M$ if and only if there is a submodule $N\subset M$ such that $R/P \cong N$. ...
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29 views

what does 2 ideals are equal mean?

I'm revisiting the proof of 1-1 correspondence theorem and while proving $f$ is one-one I don't know how to write mathematically what we mean by 2 ideals are equal? (Here $f$ is a map from set of ...
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Ideals of a skew polynomial ring where no positive power of the automorphism is inner

The exercise I'm trying to answer is as follows: Let $R$ be a ring, and $\alpha : R \rightarrow R$ an automorphism of $R$. Suppose that $R$ is simple and that no positive power of $\alpha$ is ...
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1-1 correspondence theorem

Here is the correspondence theorem stated as follows: Let $A$ be an Ideal of ring $R$.There is 1-1 correspondence between Ideals of $B$ containing $A$ and ideals of $R/A$. I have read the proof but ...
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Generators of an ideal in rings of power series

Please help me for solving a homework. Let $k$ be a field and $R=k[[x_1,x_2,\ldots,x_n]]$ the ring of power series over $k$. If $I$ is an ideal of $R$ such that ...
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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$
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Relation between ideals in Noetherian domains.

Suppose that we have a Noetherian domain $R$ and two ideals $I$ and $J$ of $R.$ Now consider the minimal (or irredundant) primary decompositions $I=\bigcap\limits_{i=1}^r Q_i$ and ...
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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 ...
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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 ...
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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 = ...
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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 ...
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Every right principal ideal non-emptily intersects the center — what is that?

This is a follow-up to Do Lipschitz/Hurwitz quaternions satisfy the Ore condition? Jyrki Lahtonen answered the question in the positive by noticing that every right principal ideal in either ring has ...
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Sets of prime ideal contain a minimal element

I want to prove that every nonempty set of prime ideal contain a minimal element, my attempt is to prove it by using zorns lemma and i would like to know if my proof is valid. Let $\Sigma$ be a ...
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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 ...
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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 $I$ ...
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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 ...
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Radical ideals of $\mathbb{Z}$?

I am having trouble with classification of the radical ideals of $\mathbb{Z}$. We know that for a commutative ring $R$ with an ideal $I$, the radical of $I$ is defined (and denoted as $\sqrt{I}$) as ...
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Non Maximal Prime ideal! [duplicate]

Assume $S$ to be all continuous functions from $[0,1]$ to $\mathbb R$. I know by compactness of $[0,1]$ it follows that all maximal ideals of $S$ have the form $M_{x_0}=\{f\in S \mid f(x_0)=0\}$.Does ...
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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 ...
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Improved Gröbner basis algorithm

I'm just learning about Gröbner bases and the Buchberger algorithm. I have seen chapters in several pieces of literature that deal with improving the Buchberger algorithm, but they never seem to ...
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a question on equivalence classes of balanced fractional ideals and Dedekind domain

Let $R$ be a commutative ring, and let $K=R\otimes \mathbb{Q}$. Def.1) We say that a pair of fractional ideals $(I, I')$ in $K$ is balanced if $II'\subseteq R$ and $N(I)N(I')=1$. Def.2) Two ...
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Preimage of maximal ideal is maximal [duplicate]

I've just started a commutative algebra course and I'm stuck on the very first homework problem: Let $A \not= \{0\}$ be a commutative ring. Let $\Phi : A \longrightarrow B$ be a ring homomorphism ...
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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 ...
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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 ...
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Let $I=(2,X)$ and $J=(3,X)$ be ideals of $\mathbb{Z}[X]$. Prove $V=\{i \cdot j : i \in I, j \in J\}$ is not an ideal. [closed]

Let $I=(2,X)$ and $J=(3,X)$ be ideals of $\mathbb{Z}[X]$. Prove $V=\{i \cdot j : i \in I, j \in J\}$ is not an ideal. Possible strategy: find $v_1,v_2 \in V$ such that $v_1+v_2 \notin V$.
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An error in the book “noncommutative ring” writed by Herstein

I'm reading the book "noncommutative ring" writed by Herstein. In the page 15, the author says that Let $F$ be a field and $A$ is an algebra over $F$. Let $\rho$ be a maximal regular right ideal ...
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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 ...
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Are ideals necessarily definable?

Consider the first-order language of rings. Let $R$ be a ring and $I \subseteq R$ be an ideal. Is $I$ necessarily $\emptyset$-definable? If not, what if we allow parameters from $R$?
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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 ...
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Equality of ideals and their vareties.

Let $I_1 $,$I_2 $ $\in \mathbb{C}[x_1,x_2,...,x_n] $ be two polynomial ideals. If their affine varieties, $\mathbb{V}(I_1)=\mathbb{V}(I_2)$ are equal then is $I_1=I_2$ always?
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non-principal height one primes of a particular hypersurface

I was reading about divisor class groups, and I was wondering the following. Let $R=\mathbb{C}[X,Y,Z,W]/(XZ-YW)$, and let $x,y,z,w$ be the images of $X,Y,Z,W$ in $R$, respectively. Is there a way ...
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63 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 ...
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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: ...
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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 $$IJ=\sum_{i,j}^{k}f_{i}g_{j}$$ ...
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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 ...
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Are there any commutative rings in which no nonzero prime ideal is finitely generated?

Are there any commutative rings in which no nonzero prime ideal is finitely generated? I feel like the example (or proof of impossibility) ought to be obvious, but I'm not seeing it.
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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 ...
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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 ...
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Ideal generated by a regular sequence

I need to prove that the ideal $$ I = (xz -y^2, x^2t^2 -yz^3, x^2yt^2 -xz^4) \subset R = \mathbb{K}[x,y,z,t]$$ is generated by a $R$-regular sequence. How can I do it? I don't know if this can ...
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Does there exist an ideal in $\mathbb{Z}_4[x]$ which is prime but not maximal?

Question: Does there exist an ideal in $\mathbb{Z}_4[x]$ which is prime but not maximal? Thoughts: It seems to me that the ideal $(x)$ fails to be a prime ideal since $0 \in (x)= 2 * 2$ with $2 ...
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61 views

Prove that $I$ is a maximal ideal of $\mathcal A$. [duplicate]

Please, give-me a hint to prove this proposition: Let $\mathcal A$ be the ring of all continuous real functions (with the usual operations of sum and multiplication) defined on the interval ...
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Functorial approach to Ideals and Quotients, Multiplicative Sets and Localizations

I have been playing with substructures of commutative rings today and noticed that there is a strong analogy between the formation of quotients and kernels with the formation of localizations with ...
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37 views

Ideal quotient and extension

Let $R$ be a commutative ring and $S$ a subring of $R$. If $I$ is an ideal of $S$ define $I^e$ as the ideal in $R$ generated by $I$, i.e. the extension of $I$ in $R$. If $I,J$ are ideals in $S$, we ...
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In $\mathbb{Z}/(n)$, does $(a) = (b)$ imply that $a$ and $b$ are associates?

[Update: Based on the hints provided by @zcn and @whacka, I believe I have found a solution. See my answer below.] Below, $R$ is a commutative ring with $1$. In John J. Watkins' Topics in ...
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Preimages of coprime ideals

Assume $R,S$ are commutative rings, $f:R\to S$ is a surjective ring homomorphism and $I,J$ are coprime ideals in $S$. Must $f^{-1}(I)$ and $f^{-1}(J)$ be coprime in $R$?