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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Interpretation of $S$-ideal class group

I have a problem understanding the interpretation of the ideal class group in the case of restricted ramifiction. Let $k$ be a number field and $S$ a set of primes of $k$. Then $k_S$ denotes the ...
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Ideal contained in the union of two ideals and a prime

Taken from Miles Reid "Undergraduate Commutative Algebra" p.35 ex. 1.12 b) Let $I,J_1,J_2 \subset A$ be ideals of a commutative ring $A$. Let $P$ be a prime ideal, then if $I \subset J_1 \cup J_2 ...
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On one-dimensional socles

Let $(R,m,k)$ be a regular local ring of dimension $n$. Let $b_1,\dots,b_n$ be a maximal $R$-sequence and define $J=(b_1,\dots,b_n)$. Let $y_1,\dots,y_n$ be a regular system of parameters of $R$ and ...
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Find a maximal ideal $I$ in the ring $\mathbb{Z}[i]$ such that $\mathbb{Z}[i]/I$ is isomorphic to $\mathbb{Z}/521\mathbb{Z}$.

I know $\mathbb{Z}[i]$, the Gaussian integers, is a PID. So $I$ is generated by a single element. At first I thought $I=(521)$, but $521$ can be reduced to $11^2 + 20^2$. Would $I=(11 + 20i)$ or ...
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simple question regarding isomorphisms relating to ring of integers

I have a simple question about isomorphisms and ideals. Let $\mathcal O_F$ be the ring of integers in some quadratic number field $F=\mathbb{Q}(\sqrt d)$ and let $f(x)$ be the minimal polynomial of ...
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Ideals of set of functions from real to real

I'm looking to prove the following is an ideal of the set of functions from real numbers to real: a)the set of all f such that f(x) = 0 for every rational x b) the set of all f such that f(0) = 0
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Smallest skew-field containing a non-commutative ring.

Let $R$ be an integral domain and take $D = R - \left\{ 0 \right\}$. The ring $D^{-1}R$ is the smallest field containing $R$ as a subring. Now suppose that I have a non-commutative ring $N$. Suppose ...
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A question on valuation overrings of a PID

Let $A$ be a PID and let $K$ be its quotient field. Let $V$ be a valuation ring of $K$ containing $A$ and assume $V\neq K$. Show that $V$ is a local ring $A_{(p)}$ for some prime element $p$. I ...
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47 views

Multiplicative identity of a quotient ring

Let $R$ be a commutative ring such that $M$ is a maximal ideal of $R$. Then I know that $R/M$ is a field. But I am unable to understand what is the multiplicative identity of $R/M$? If $R$ has an ...
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Maximal ideals of rings which are finite dimensional vector spaces over $\mathbb C$.

If $K$ is a commutative ring which is a finite dimensional vector space over $\mathbb C$ what can we say about the maximal ideals of $K$? What can we say if instead of $\mathbb C$ we have some ...
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Adjoin to a ring an element satisfying a monic vs. non-monic polynomial.

Artin defines the adjoining of an element $\alpha$ to ring $R$ satisfying polynomial $f \in R[x]$ by $$R[\alpha] = R[x]/(f).$$ If $f$ is monic with degree $n$, then we get some nice properties, such ...
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How many elements are there in this quotient ring?

So we are having this undergraduate course in my department of commutative algebra and there is a problem sheet that we have to submit. The second problem goes like this: Let $R$ be the ring ...
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63 views

Ideals of $R/N$

I have a proof in a book I'm reading that says (we are dealing with ring $R$ and ideal $N$) that the ideals of $R/N$ are of the form $M/N$ where $M$ is an ideal of $R$ that contains $N$. Can someone ...
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Non-zero prime ideals of $F[x]$ are maximal

Prove that if $F$ is a field, every proper prime ideal of $F[X]$ is maximal. Should I be using the theorem that says an ideal $M$ of a commutative ring $R$ is maximal iff $R/M$ is a field? Any ...
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118 views

About proving that maximal ideals are prime

Let $R$ be a Ring with unity, and $I$ a maximal ideal in $R$. Show that $I$ is a prime ideal. I have seen this proof in many places If $ab\in I$ and $a\notin I$, then $I+(a)=R$ and hence there is ...
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Determining if ___ is an ideal:

I am reading some notes on algebra, with polynomial rings and there is an exercise that asks to determine if 1-5 are ideals. I am not very familiar and am hoping that the examples will give a more ...
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Polynomial Ideal: x+y element of an ideal implies what?

If x+y $\in I$, where I is a subset of the polynomial ring and is an Ideal, can we say that $x \in I$? Edit: in particular, let I= $\langle x^2, x+y \rangle$. Then this is equal to $\langle y^2, x+y ...
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Endomorphism of a local $k$-algebra inducing an automorphism modulo $m^2$ is an automorphism

The following is exercise 4.1 of Hartshorne's Deformation Theory, used in the proof given there of the sufficiency of the infinitesimal lifting criterion of smoothness: Let $(A,m)$ be a local ...
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Intersection and Sum of Polynomial Ideals from different rings

It is well known that intersection and sum of polynomial ideals from the same ring are lattice operations. I wonder if this is still true for ideals from different rings (over the same field). ...
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Calculating Grobner Bases of Subideals

Suppose we have an ideal $I = (f_1,\dots,f_k) \subset \mathbb{Z}[x_1, \dots, x_n]$. Let $J = (f_1, \dots, f_s) \subset I$. Suppose we knew that $I$ and $J$'s Grobner bases, under the standard ...
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Decide belonging of an element to an ideal

Let $f_1=x^2-y, f_2=xy-z,f_4=xz-y^2$ be polynomials with coefficients in some field $k$. I want to prove that $f_2\notin (f_1,f_4)$. My attempt: by contraddiction, let $f_2\in (f_1,f_4)$. Then there ...
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Quotient rings in formal power series

I'm trying to find all prime ideals in the formal power series ring $k[[x]]$, where $k$ is a field. I think I've managed to show that all ideals are of the form $<x^n>$, $n>0$, i.e. ...
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Ideal with large Grobner basis with respect to one monomial order

What is an example of a set of at most four polynomials $f_1,\ldots,f_n$ (in any number of variables) such that $\{f_i\}$ is a Grobner basis of $I=\langle f_i\rangle$ with respect to one monomial ...
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I as an ideal of $R$ then $a+I=0+I$ iff $a\in I$ [duplicate]

show that if a,b belong to the ring $R$ and $I$ is an ideal of $R$ then $a+I=0+I$ if and only if $a$ belongs to $I$. I know that since I is an ideal then it is both a left and a right ideal.
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Maximal ideal of $\Bbb Z$ that is not maximal in $\Bbb Z[X]$

Can someone come up with an example of a maximal ideal P in $\mathbb{Z}$ such that P[X] is not maximal in $\mathbb{Z}[X]$ - the ring of polynomials with integer coefficients? I know that the maximal ...
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180 views

Prove that all ideals in Q[x] are principal

Prove that all ideals in the polynomial ring $\mathbb{Q}[x]$ are principal. There is probably some elegant shortcut one can use for this proof, but I am only just beginning to study ring theory and ...
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79 views

Ideals of Lattice.

If $L$ is a lattice then the ideal $I$ of $L$ is a nonempty lower segment closed under join. I need to show that the set of ideals $I(L)$ of $L$ forms a lattice under $\subseteq$ I know the ...
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Trying to prove pre image of product of ideals is, the product of the pre images of the two ideals…

This is from Elements of Abstract Algebra by Allan Clark. 166 $\beta$ $\Phi ^{-1}(a'b') = (\phi^{-1}(a'))(\phi^{-1}(b'))$ I can prove an element of the right side is an element of the left. But I ...
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Intersection of Two Ideals in the Ring of Integers

I'm working through this question If the intersection of $x\mathbb{Z}$ and $y\mathbb{Z}$ equals $w\mathbb{Z}$ for positive integers $x$, $y$, and $w$ (ideals in ring $\mathbb{Z}$), express the ...
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The maximal ideal in a local ring is finitely generated

Assume $m<R$ is the maximal ideal of a commutative local ring with identity, such that $m=m^2$. Is $m$ finitely generated? Is the condition $m=m^2$ redundant? I am trying to apply Nakayama's lemma ...
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Embedding dimension one problem

Let $(R,m)$ be a local noetherian ring and $\dim R=0$. If $\dim_km/m^2=1$ a) show that $m$ is a principal ideal. b) prove that every nonzero ideal is parametric and principal. (An ideal $q$ called ...
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Checking if something is a unit

Check if $\mathbb{Z}_5/x^2 + 3x + 1$ is a field. Is $(x+2)$ a unit, if so calculate its inverse. I would say that this quotient ring is not a field, because $<x^2 + 3x + 1>$ is not a maximal ...
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Rings, annihilators and (maximal) ideals

Let $R$ be a unital, associative, non-commutative ring. If $P$ is an ideal of $R$, what is the annihilator of quotients $R/PR$ and of $R/P$? Does something change if $P$ is supposed to be a maximal ...
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Union of ascending ideals is an ideal

Could you tell me what I'm doing wrong in proving this proposition? If $I_1 \subset I_2 \subset ... \subset I_n \subset ...$ is an ascending chain of ideals in $R$, then $I := \bigcup _{n \in ...
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Kernel and direct sum

Let $R=k[x_1,\ldots,x_7]$ be a polynomial ring over field $k$ and $I=\bigcap_{i=1}^4 \mathfrak{p}_i$ where $\mathfrak{p}_1=(x_1,x_3,x_5,x_6), \mathfrak{p}_2=(x_1,x_3,x_4,x_6), ...
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Computing kernel

Let $I,J$ be two ideals of Noetherian ring $R$. How to compute kernel of following homomorphism directly: $$\phi: R/I\oplus R/J\to R/(I+J) $$ $$(a+I,b+J)\to (a-b)+I+J $$
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The kernel of homomorphism of a local ring into a field is its maximal ideal?

I have a question about the proof of Theorem 3.2. of Algebra by Serge Lang. In the theorem $A$ is a subring of a field $K$ and $\phi:A \rightarrow L$ is a homomorphism of $A$ into an algebraically ...
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Ideal for two polynomials in three variables

Consider the set $B=\{(t^2,t^3,t^4)\mid t\in \mathbb{C}\}$. It is a subvariety of $\mathbb{C}^3$, because it is equal to $V(y^2-x^3,z-x^2)$. How can we find the ideal $I(B)$? I think it is $I(\langle ...
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Maximal Ideals in the Integers

We know that the maximal ideals in $\mathbb{Z}$ are all ideals of the form $(p)$, where $p$ is prime. But what if we consider $(p, p_2)$, where $p_2$ is prime, although I'm not sure that the ...
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Ideal of variety in polynomial ring

Let $k$ be a field, let $A$ be an ideal of $k[x_1,\ldots,x_n]$, and let $B$ be an ideal of elements $f\in k[x_1,\ldots,x_n]$ such that $f^n\in A$ for some $n$. I want to show that $B\subseteq ...
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How to prove that $J(M_n(R))=M_n(J(R))$?

How to prove that $J(M_n(R))=M_n(J(R))$? Here $M_n(R)$ is the ring of matrices of size $n^2$ over the ring $R$. And $J(M_n(R))$ is a two-sided ideal of the ring $M_n(R)$.
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$(\mathfrak{a}+\mathfrak{b}:f)\stackrel{?}{=}(\mathfrak{a}:f)+(\mathfrak{b}:f)$

Suppose $R$ is a commutative ring with unit and $\mathfrak{a},\mathfrak{b}$ are ideals of $R$. We can define $(\mathfrak{a}:f)=\{g\in R\mid gf\in \mathfrak{a}\}$ for $f\in R$. I can see that ...
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Showing 3 is an irreducible element of $\mathbb Z[\sqrt{2}]$

What I tried: $$(3)=\{3r|r\in \mathbb Z\}\space\mbox{is a maximal ideal of}\space\mathbb Z\implies(3)=\{3(a+b\sqrt{2})|a,b\in\mathbb Z\}\space\mbox{is a maximal ideal of }\mathbb ...
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A prime is minimal among primes containing an ideal

Let $I$ be an ideal in a noetherian ring $R$, and $P$ prime containing $I$. I must prove that if in the localization $R_P$, $R_P/I_P$ is annihilated by a power of $P_P$, then $P$ is minimal among ...
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factorization of ideals

Let $L$ and $K$ be number fields such that $L/K$ is a finite extension. Suppose $\mathfrak{a},\mathfrak{b}$ are ideals in $\mathcal{O}_K$ and $\mathfrak{a}\mathcal{O}_L|\mathfrak{b}\mathcal{O}_L$. ...
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In the ring $R$ of real-valued functions on $(0,1)$, $I = \{f \in R \mid f(1/2)=0\text{ and }f(1/3)=0\}$ is not a prime ideal

So I already proved that $I$ is an ordinary ideal, now it's time to prove that it's not a prime ideal. So I need to show that $R/P$ is not an integral domain, but I'm confused as to how to go about ...
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Is the universal enveloping algebra of the free Poisson algebra generated by finite set (left) noetherian?

Let $P$ be the free Poisson algebra over $k$ (a field) generated by finite set $x_1,...,x_n$. Let's consider the universal enveloping algebra $P^e$ of the free Poisson algebra $P$. Hence a Poisson ...
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Clarification about some proof of Projectivity

Small provides an example of a ring which is right but not left hereditary is the ring $R =\left( \begin{matrix} \mathbb{Z} & \mathbb{Q} \\ 0 & \mathbb{Q} \end{matrix} \right)$; To ...
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94 views

Identifying some quotient rings

How come that $k[w,z]/(w^2+z,w^3 z^2)\cong k[w]/(w^7)$? Also why is $(xz,w)=(x,w)\cap(z,w)$ in the polynomial ring in 3 variables? what are the rules of ideal calculus making these results evident?
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Two ideals both alike in dignity, in fair Paris where we lay our scene. (proving ideals are isomorphic)

Let $A$ be an integral domain. I have to show that two ideals $\mathfrak a$ and $\mathfrak b$ are isomorphic as $A$-modules if and only if there exist $a$ and $b$ such that $a\mathfrak b=b\mathfrak ...