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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Finding homomorphisms and kernels from a given ring R

Give the following rings $R$ and ideals $I$ find a ring $S$ and a homomorphism $f:R \rightarrow S$ with kernel $I$ i) $R=\mathbb{Q} [x], I=(x^{2}-2)R$ ii)$R=\mathbb{Z}[i], I=2R$ (Gaussian Integers) ...
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Does $A \Delta N = A \cup N'$ in this proof?

Please have a look at this topic: $\sigma$- ideal The answer says: $A\Delta N=A\cup N'$ with $N'=N\setminus A$. I do not see why this is correct.. Usually it is $A\Delta N=(A\setminus N)\cup ...
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Unique maximal ideal implies set of non-units is an ideal

This is not for homework, but I would just like a hint please. The question asks If a commutative ring $R$ (with $1$) has a unique maximal ideal, then the set of non-units in $R$ is an ideal. ...
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$\sigma$- ideal

Let $(\Omega,\mathcal{A})$ be a measurable space. $\mathcal{N}\subset\mathcal{P}(\Omega)$ is called a $\sigma$ ideal, if $$ (1)~\emptyset\in\mathcal{N},~~~~~(2) N\in\mathcal{N}, M\subset ...
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55 views

Painting $\mathbb R^+$ with two colors which sum of two same color numbers be the same.

Can any one paint $\mathbb R^+$ with two colors which sum of two numbers with the same color has the same color. Additional condition: Both colors should be used. I tried use Cauchy functions like ...
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42 views

Mistake in the proof that a domain is flat as a module over any subring

Where is the mistake in the following argument? I feel that there has to be one, for example by the very existence of this article. Let $R$ be an integral domain and $S \subseteq R$ be a subring ...
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57 views

Are these ideals the same?

I have already proved that $(X^3-Y^3,X^2Y-X)\subseteq(X^2-Y,X-Y^2)$ since the elements $X^3-Y^3$ and $X^2Y-X $ can be written as a linear combination of $(X^2-Y,X-Y^2)$. However, I can't write ...
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60 views

Proving prime ideal

Prove that $P$ is prime if and only if it has this property: Whenever $A$ and $B$ are ideals in $R$ such that $AB \subseteq P$, then $A \subseteq P$ or $B \subseteq P$, where $P$ be an ideal in a ...
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185 views

Describing the ideal in polynomial ring with $n$ indeterminates

Let $R$ be a commutative ring and let $m$ be a natural number. Describe the ideal $(X_1,X_2,...,X_n)^m$ of the ring $R[X_1,X_2,...,X_n]$ of polynomials over $R$ in indeterminates $X_1,...,X_n$. I ...
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Class group and factorizations

There is a common characterization of the class group ${\rm Cl}(R)$ as a kind of measure of how badly factorization fails to be unique. The most obvious justification for this sentiment is that the ...
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a principal ideal contains a monic polynomial of least degree n

Q11.3.11 Artin Algebra 2nd Let $R$ be a ring, and let $I$ be an ideal of the polynomial ring $R[x]$. Let $n$ be the lowest degree among nonzero elements of $I$. Prove or disprove the ...
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Ring homomorphism homework including ideals and surjectivity.

$R$ is a ring and $I$ and $J$ are ideals of $r$. Show that the ring homomorphism $h:R \rightarrow R/I \times R/J, r \mapsto (r+I,r+J)$ is surjective iff $I+J=R$ give a description of the kernel of ...
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If a field contains the complex field, then it is $\mathbb{C}$

This question is originated from a book by Gaal, (Linear Analysis and Representation Theory). Theorem 7 from section 6, chapter 1 reads as follows, and I quote: "Theorem 7: Let $A$ be a complex, ...
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Show that quotient ring of a $\Bbb C$-algebra by a maximal ideal is isomorphic to $\mathbb{C}$.

Let $R = \mathbb{C}[x_1,...,x_n]/I$ be a quotient of a polynomial ring over $\mathbb{C}$, and let $M$ be a maximal ideal of $R$. How do I show that quotient ring $R/M$ is isomorphic to ...
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24 views

Ideals in a Noetherian ring

Let $R$ be a ring, let $\mathfrak{i}$ be an ideal of $R$, let $\{x_{\alpha}\}_{\alpha\in\mathcal{A}}$ be a set of generators for $\mathfrak{i}$, suppose $\mathcal{A}$ has infinitely many elements. ...
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32 views

In $\mathbb{Z}_5[x]$, let $I=\langle x^2+x+2\rangle$. Find the inverse of $2x+3+I$ in $\mathbb{Z}_5[x]$.

In $\mathbb{Z}_5[x]$, let $I$ be the ideal generated by $x^2+x+2$, $I=\langle x^2+x+2\rangle$. Find the inverse of $2x+3+I$ in $\mathbb{Z}_5[x]$. I don't understand what $2x+3+I$ means. Is it just ...
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Prove that $I= \{a+bi \in ℤ[i] : a≡b \pmod{2}\}$ is an maximal ideal of $ℤ[i]$.

I'm making some exercises to prepare for my ring theory exam: Prove that $I= \{a+bi \in ℤ[i] : a≡b \pmod{2}\}$ is an maximal ideal of $ℤ[i]$. I know that $a+2l=b$ with $l\in ℤ$ (or should I ...
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108 views

Let $I$ be a prime ideal. Show that $f^{-1}(I)$ is a prime ideal of $R$. Is this also true for maximal ideals?

I'm making some exercises to prepare for my ring theory exam. Let $f:R→R'$ be a ring homomorphism, with $f(1)=1$, and $R,R'$ commutative rings with $1$. Let $I$ be a prime ideal. Show that ...
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Augmenting «$\Bbb Z[x]$ f.g. $\Rightarrow x$ integral» for ${\frak p}[x]$

In KCd's blurb on ideal factorization, page 5: $\hskip 0.3in$ The situation is this: $K$ is a number field, ${\cal O}_K$ its ring of integers, ${\frak p}\triangleleft{\cal O}_K$ a prime ideal, ...
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showing every ideal of some quotient ring is principal.

Let $\mathbb F$ be a field and $A=\mathbb F[t]/(t^2)$, where $(t^2)$ is the ideal of $\mathbb F[t]$ (This quotient ring is not an integral domain as you know), and I write an element of $A$ by ...
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Prove that the elements $1, t-a, (t-a)^2, (t-a)^3,\dots, (t-a)^{n-1}$ form a $\mathbb{C}$-basis for the quotient ring $\mathbb{C}[t]/((t-a)^n)$.

Prove that the elements $1, t-a, (t-a)^2, (t-a)^3,\dots, (t-a)^{n-1}$ form a $\mathbb{C}$-basis for the quotient ring $\mathbb{C}[t]/((t-a)^n)$. $((t-a)^n)$ is the ideal generated by $(t-a)^n$. ...
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Prime ideals in $C[0,1]$

Are there any prime ideals in the ring $C[0,1]$ of continuous functions $[0,1]\rightarrow \mathbb{R}$, which are not maximal? Perhaps, I duplicate smb's question, but this is an interesting problem! ...
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Non-modular maximal ideal in abelian Banach algebra

Let $A$ be the disk algebra (i.e. the algebra of all functions that are continuous on the closed unit disk and analytic on the open unit disk) and let $A_{0}=\{f\in A:f(0)=0\}$. Then $A_{0}$ is a ...
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Let R be a commutative ring and let $A$ and $B$ be ideals of $R$. Show that if $A + B = R$, then $AB = A\cap B$

Let $R$ be a commutative ring and let $A$ and $B$ be ideals of $R$. (i) Show that if $A + B = R$, then $AB = A\cap B$ (ii) Now let $R$ be a Euclidean domain. prove that if $AB = A\cap B$, then $A + ...
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semigroup ideals

For the semigroup $$‎S_{3 \times 3} = \bigl\{(a_{ij}) \bigm| a_{ij} \in \mathbb Z_2 = \{0,1\}\bigr\}$$ ‎‎(the set of all $‎3‎\times‎‎3$ matrices with entries from $\mathbb Z_2$) under multiplication. ...
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Defining an ideal in the tensor algebra

In the wikipedia article about exterior algebra: The exterior algebra $Λ(V)$ over a vector space $V$ over a field $K$ is defined as the Quotient algebra of the tensor algebra by the two-sided ...
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Proving that $\{f \in End(A): \forall a \in A:|a|<\infty \implies f(a)=0\}$ is an ideal

I need to prove that $I = \{f \in End(A): f(a)= 0 \ \text{for all $a \in A$ with a finite order}\}$ It isn't hard to prove that $I$ is a subgroup of $End(A)$, but it is quite hard to prove that: ...
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Generating set for a polynomial ideal

I would like to know which is the generator set for the following polynomial ideal: $$ I=\{a_nx^n+\cdots +a_0\in\mathbb{Z}[x]\,\, | \,\, a_0\,\, \text{is even}\}. $$ Sorry for the writing.
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Ideals in a ring satisfying $I \cap (J+K) \neq I \cap J + I \cap K$

I am not getting an example where $I \cap (J+K) \neq I \cap J + I \cap K$, where $I,J,K$ are ideals in a ring $A$.
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Are these definitions of a prime ideal equivalent?

I just noticed I have three different definitions of a prime ideal in my notes. So are these definitions equivalent? Are they all correct...I have feeling I might have taken something down wrong. Let ...
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What is the field of definition of an invariant ideal?

Let $K/k$ be a finitely generated field extension, such that $k=K^G$ for some (possibly infinite) set $G$ of automorphisms of $K$. Now, consider the extension of polynomial rings $$ ...
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Nilradical of $\mathbb{R}[X,Y]/(X^nY^m)$

I'm having trouble finding the nilradical of $\mathbb{R}[X,Y]/(X^nY^m)$ for given $n$ and $m$. I believe the nilradical is $\{f(XY) \in \mathbb{R}[XY] : f \textrm{ has constant term 0}\}/(X^nY^m)\}$. ...
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Example of a commutative ring with identity with two ideals whose product is not equal to their intersection

I need a specific example of a commutative ring with identity, and two ideals in the ring whose product is not equal to their intersection. I know that for two such ideals I and J, IJ = I ∩ J if I + ...
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Prove : If $I = (p(x))$ is a prime ideal in $F[x]$ then $p(x)$ is irreducible.

I have to show : If $I = (p(x))$ is a prime ideal in $F[x]$, where F is a field, then $p(x)$ is irreducible. In the book I use, there is the proof of the converse which uses Euclid's Lemma. I ...
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Polynomial quotient ring : k[x,y,z,t]/(xy-zt)

I have some trouble picturing a quotient. Namely, what $k[x,y,z,t]/(xy-zt)$ looks like where $k$ is a field ? My intuition is probably wrong but is it isomorphic to $k[u,v,w]$ ?
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How to compute the product of ideals

Let $R=\mathbb{Z}[\sqrt{-105}]$. Consider the following ideals in $R$: $\mathfrak{p}_2=(2,1+\sqrt{-105})$ $\mathfrak{p}_5=(5,\sqrt{-105})$ $\mathfrak{p}_{13}=(13,5+\sqrt{-105})$ I know the ...
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Maximal ideal of Cauchy nullsequences

Let $(K, \left|\phantom{x}\right|)$ be a valued field. Let $\mathscr{C}$ be the commutative ring with unity given by Cauchy sequences of elements in $K$, with $+$ and $\cdot$ defined term by term. ...
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$(\bigoplus_{A}M_{\alpha })/ I(\bigoplus_{A}M_{\alpha }) \cong \bigoplus _{A}M_{\alpha }/IM_{\alpha }$

Let $(M_\alpha )_{\alpha \in A}$ be an indexed set of left R-modules and let $I$ be a left ideal of $R$. Prove that $I(\bigoplus_{A}M_{\alpha })= \bigoplus_{A} IM_{\alpha }$ and ...
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$tI$ is an ideal in $F[t, x_{1}, x_{2}, …, x_{n}]$ for every ideal $I$

The question I want to ask seems very clear for everybody. In fact it is a point in the proof of how to compute intersection of 2 finitely generated ideal, but I don't think it's true, or maybe I ...
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prime ideals of $\mathbb{Z} / 36\mathbb{Z}$

Could I have a good explanation about why the prime ideals of $\mathbb{Z}/36\mathbb{Z}$ are $2\ \mathbb{Z}$ and $3\ \mathbb{Z}$?
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Example showing that the product of ideals must be the span of the commutators

I'm trying to find an example showing why, in a Lie algebra, we can't just define the product of two ideals $I$ and $J$ to be the elements of the form $[x,y]$ where $x \in I, \; y \in J$. I imagine ...
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property of prime ideals

Let $P$ be a prime ideal of $R$. I want to prove that if $J$ and $K$ are left ideals of $R$ such that $JK\subseteq P$, then either $J\subseteq P$ or $K\subseteq P$. I think I want to try something ...
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Showing that the ideal of matrices with entries from an ideal is maximal iff that ideal is maximal

I know that if $J$ is an ideal in a ring $R$ then $M_n(J)$, the set of all $n\times n$ matrices with entries in $J$, is also an ideal. How would I show that $M_n(J)$ is maximal iff $J$ is maximal? I ...
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Maximal ideal in the ring of functions from $\mathbb{R} \to \mathbb{R}$

Well, the problem I'm trying to solve is this: Let A be the ring of all continuos functions from $\mathbb{R} \to \mathbb{R}$. Show that $$M = \{f \in A; f(0)=0\}$$ is a maximal ideal of A. I tried ...
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Let's cover-rings.

Let $R$ be a (commutative) ring with identity. A covering of R is a subset $\{r_1,...,r_n\}$ of elements of $R$ such that $R$ is generated by $r_1,...,r_n$. I think that if $\{r_1,...,r_n\}$ is a ...
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Prove that ring contains infinitely many minimal prime ideals

I get stucked on this problem, hope some one can help me solve this. Prove that the ring $\mathbb Z[x_{1}, x_{2}, ...]/(x_{1}x_{2}, x_{3}x_{4},x_{5}x_{6}, ...)$ contains infinitely many minimal ...
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Ideals in infinitely variables polynomial are not finitely generated

I'm doing this exercise in the Dummit and Foote textbook and got no clue for it. Hope some one can help me solve this. Thanks Prove that a polynomial ring in a infinitely many variables with ...
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351 views

Show that ideal is a subring

I'm experimenting around with ring ideals (perhaps ideals is always for rings, so when speaking of ideals we always refer to these ring subsets?), and my book gives me the definition that an ideal $I$ ...
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130 views

Is $(x^2+y^2-1,z-iy)$ a prime ideal in $\mathbb C[x,y,z]$?

Is $(x^2+y^2-1,z-iy)$ a prime ideal in $\mathbb C[x,y,z]$? How can I prove it? I need this to decompose the algebraic set $V(x^2+y^2-1,x^2-z^2-1)$ into irreductible components.
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Is $\operatorname{Spec}(\mathbb{C}[x,y]/(y-x^2))$ the same as $\operatorname{Spec}(\mathbb{C}[x])$ ?

Let $R=\mathbb{C}[x,y]/(y-x^2)$. We have that $$\operatorname{Spec}(R)=\{(0),(x-a,y-a^2),(y-x^2)\}. $$ But if we consider the quotient ring $\mathbb{C}[x,y]/(y-x^2) \simeq \mathbb{C}[x]$. But ...