This tag is for questions about rings, which are a type of algebraic structure studied in abstract algebra and algebraic number theory.

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1answer
69 views

Help showing $\frac{\mathbf{C}[x]}{(x-1, x-2)} \cong \mathbf{C}^2$ (EDIT: FALSE)

My algebra professor told us as an exercise that $$\frac{\mathbf{C}[x]}{(x-1, x-2)} \cong \mathbf{C}^2. $$ I have been having trouble showing this is true. Can anyone help me out? Thanks
4
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2answers
47 views

Understanding of an example of “extending scalars”

The following is an example in the Abstract Algebra by Dummit and Foote: I don't understand in this example why $\iota$ is an isomorphism. By Theorem 8, I can get $$ id_N=\Phi\circ\iota $$ which ...
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3answers
35 views

Is $f(a)=0^{\prime}$ bijective homomorphism?

Let $f:(R,+,.)\rightarrow (R^{\prime},+^{\prime},.^{\prime})$, and $f(a)=0^{\prime}$, $\forall a \in R$. Is $f$ bijective homomorphism? My answer is no because $f$ is not injective function so its ...
2
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1answer
26 views

What are the primes in quadratic integer ring $\mathbb{Z}[D]$

Is it possible to classify all the primes in the ring of integers $\mathbb{Z}[D]\subseteq\mathbb{Q}(\sqrt{d})$? If not, are there additional assumptions on $d$ like residue mod 4 or the "class number" ...
0
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1answer
35 views

Example of Artinian module which has infinitely many maximal submodules not isomorphic to each other

I'm looking for an Artinian module which has infinitely many maximal submodules not isomorphic to each other. I guess I can find it over a matrix ring.
2
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1answer
19 views

Is every element in $(P(X),\cap,\cup)$ a zero divisor?

Let $X$ be a non empty set. Is every element in $(P(X),\cap,\cup)$ a zero divisor?
5
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1answer
34 views

Is $(\mathbb{Z}_{n},+_{n},._{n})$ a field, $\forall n\in \mathbb{N}$?

Is $(\mathbb{Z},+_{n},._{n})$ a field, $\forall n\in \mathbb{N}$? My answer is No, because for $n=6$, $(\mathbb{Z}_{6},+_{6},._{6})$ has a zero divisor but a field has no zero divisors so it can't be ...
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1answer
35 views

Difference between irreducible factors in $\mathbb{Z}[X]$ and $\mathbb{Q}[X]$

I don't fully understand the difference between factorizing a polynomial in irreducible factors in $\mathbb{Z}[X]$ and $\mathbb{Q}[X]$. For example $f=X^4-X^2+4X+3$ is irreducible in $\mathbb{Z}[X]$, ...
1
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1answer
41 views

Factorizing $X^4-Y^2$ in $\mathbb{Q}[X,Y]$

I want to factorize $X^4-Y^2$ in $\mathbb{Q}[X,Y]$ in irreducible factors. I thought about using Eisenstein's criterium to show that it is irreducible, though I'm not sure what the prime element is ...
3
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0answers
40 views

Homomorphisms of modules over a corner ring

Let $R$ be a Noetherian ring and suppose that we can write $1 = e_1 + e_2 + \dots + e_n$ where the $e_i$ are pairwise orthogonal idempotents. Let $S = e_1 S e_1$, and consider the right $S$-modules ...
4
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1answer
48 views

Basis for $\text{Mat}_2(\mathbb{Z})$ as a $\mathbb{Z}[i]$-module

Let $M=\text{Mat}_2(\mathbb{Z})$ a $\mathbb{Z}[i]$-module with scalar multiplication ...
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3answers
169 views
+200

Is $\mathbb{Z}[\sqrt{15}]$ a UFD?

Let $R=\mathbb{Z}[\sqrt{15}]=\{a+b\sqrt{15}:a,b\in\mathbb{Z}\}$. How do I show that $(3,\sqrt{15})$ is a maximal ideal but not a principal ideal? How do I show that $(3,\sqrt{15})^2$ is a ...
0
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1answer
34 views

If a ring has no zero divisor, Is a every subring of that ring has no zero divisor? [closed]

If a ring has no zero divisor, then is the following statement true? Every subring of that ring has no zero divisor
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3answers
41 views

Is there a subring that has a different identity element from the original ring?

Let $(S,+,.)$ be some ring. Is there a subring of $(S,+,.)$ that has a different identity element from the identity element in $(S,+,.)$? I am new to ring theory so please let answers simple.
0
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1answer
49 views

Irreducible is Prime?

I know that this property holds for euclidean rings, PIDs, UFDs and GCD domains. But does it hold for Bezout rings, Noetherian rings or ACCP rings?
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2answers
47 views

a property of the tensor product of modules

The following theorem is from the Abstract Algebra by Dummit and Foote (in the section 10.4 tensor products of modules): Would anybody illustrate how Theorem 8 is used to get $$ \textrm{ker ...
0
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1answer
33 views

All local subrings of $\mathbb Q$ are valuation rings of $\mathbb Q$?

Let $R\subseteq\mathbb{Q}$ be a local subring with maximal ideal $\mathfrak{m}$. Is $R$ a valuation ring of $\mathbb Q$? $R$ is a valuation ring iff its ideals are linearly ordered. But I'm stuck ...
0
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0answers
24 views

Convolution vs term wise product

Fix an element $x\in\mathbb R^\times$. Then I can define the ring $\mathbb R[[x]]$ which is the set: $$\left\{\sum_{i=0}^\infty a_ix^i\,: a_i\in\mathbb R\right\}$$ and where the product is the usual ...
3
votes
1answer
50 views

Principal Ideals and Identity elements

When defining a principal ideal, e.g $I$ $I=aR=(ar:r \in R)$ (an ideal generated by a single element of the ring)do we require the ring to have an identity element and if so, in what manner does ...
0
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2answers
33 views

Can we always choose the generators of an ideal of a Noetherian ring to be homogeneous?

Let $R$ be a $k$-subalgebra of $S=k[x_1,x_2,\dots,x_n]$. Let $m\in R$ be the ideal generated by the homogeneous elements of $R$. As $S$ is Noetherian, the ideal $mS$ has a finite set of generators, ...
1
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2answers
61 views

Prime factorization of Gaussian integers

I want to find $a, b\in\mathbb{Z}[i]$ such that $a(2+3i)+b(5+5i)=1$. I don't know how to do this, but my first thought was to do something with the norm or otherwise factoring ...
0
votes
1answer
36 views

Determine if an ideal is prime

Let $R=\mathbb{Z}[x,y]$ and $I=(y-x^2)R+(x-4)R$ is ideal $J$ prime ? I tried to produce such $a,b$ that $ab \in J$ but $a,b \not\in J$ but can't find so far
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1answer
35 views

Irreducible polynomials in $\mathbb{C}[X,Y]$

I have the polynomial $X^2+Y^2-1$ in $\mathbb{C}[X,Y]$. Is this irreducible? If not, how do I factorize it? Should I handle this the same as if it were $\mathbb{R}[X,Y]$, or should I do it ...
3
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1answer
34 views

Show that certain matrices over rings form a field

I have got the following assignment: $R$ is a ring, $K:=\{ \begin{pmatrix} a & b \\ -b & a \\ \end{pmatrix}: a,b \in R\}$ I need to show that $K$ is a field. And I believe it is not ...
1
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1answer
46 views

For ideals $I=(8)$ and $J=(5+5i)$ in $\mathbb{Z}[i]$, what are $IJ$, $I+J$ and $I\cap J$?

Let $I=(8)$ and $J=(5+5i)$ be ideals in $\mathbb{Z}[i]$. How do I find $x,y,z\in\mathbb{Z}[i]$ such that: $IJ=(x)$, $I+J=(y)$, $I\cap J=(z)$? Is it correct that $y=13+5i$ and $x=40+40i$?
2
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1answer
32 views

How do I show that $\text{End}_R\mathbb{Z}^n\cong\mathbb{Z}$ for $R=\text{Mat}_n(\mathbb{Z})$?

Let $R=\text{Mat}_n(\mathbb{Z})$ and $M=\mathbb{Z}^n$ the (left) $R$-module with action the matrix multiplication. How do I prove that $\text{End}_RM\cong\mathbb{Z}$? Should I find an explicit ...
0
votes
4answers
25 views

Number of ring homomorphisms from $\mathbb{Z}[X]/(X^2-X)$ to $\mathbb{Z}/6\mathbb{Z}$

I want to determine the number of ring homomorphisms from $\mathbb{Z}[X]/(X^2-X)$ to $\mathbb{Z}/6\mathbb{Z}$. If we suppose that such a homomorphism is not the zero homomorphism, then ...
2
votes
1answer
28 views

Proving that a $\mathbb{Z}[i]$-module is free

I have the $\mathbb{Z}[i]$-module $$A=\{(z,w)\in\mathbb{Z}[i]^2:z+(1+i)w=0\bmod(2+2i)\}.$$ I want to prove that it is free. I know that being free means that $A$ is generated by a linearly ...
1
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1answer
36 views

Isomorphic quotients of polynomial rings over finite fields

What are the elements of $\mathbb{F}_3[X]/(X^3-3)$? A similar question was posted here: Elements of the field $F_2[x] / (x^3 + x + 1)$, but it doesn't explain why the elements of that field look ...
0
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1answer
41 views

Is there an explicit formula for the expression $(a\mathbb{Z}+b) \cap (a'\mathbb{Z}+b'),$ not involving $\cap$?

Thinking of $\mathbb{Z}$ as a ring, the ideals of $\mathbb{Z}$ are precisely those subsets of the form $a\mathbb{Z}.$ Hence intersections of ideals can be computed by taking lowest common multiples. ...
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3answers
53 views

Prove that if an integer $z$ is not divisible by $p$, then it is invertible in the $p$-adic integer ring $\mathbb{Z}_p$.

Let $p$ be a prime number. Define the $p$-adic valuation on $\mathbb{Z}$ as $v_p(p^kx) = p^{-k}$ where $x$ is not divisible by the prime $p$. Let $\mathbb{Z}_p$ (the $p$-adic integers) be a ...
0
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2answers
95 views

Are rings $\mathbb Q[i]$ and $\mathbb Q[\sqrt{3}i]$ isomorphic?

I wonder if the rings $\mathbb Q[i]=\{a+bi: a,b \in \mathbb{Q}\}$ and $\mathbb Q[\sqrt{3}i]=\{a+\sqrt{3}bi:a,b \in \mathbb{Q}\}$ are isomorphic. I tried map $f(a+bi)=a+\sqrt3bi$ but then it ...
0
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2answers
37 views

When is $R[x]$ a Domain? [closed]

Is there a connection between $R$ having zero divisors and $R[x]$ having zero divisors?
0
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3answers
93 views

Adding an inverse of $(2, 0)$ to $\Bbb{R\times R}$

This is a problem from Artin: Describe the ring obtained by adding an inverse of $(2, 0)$ to $\Bbb{R\times R}$. I can see that this also adds inverses for all elements of the form $(a, 0)$, but ...
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1answer
83 views

A special class of commutative rings

Can we characterize all commutative finite rings the sum of all elements is zero? For instance, all odd characteristic finite commutative rings have this property (easy). GF(2) does not.
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0answers
22 views

If the union of $n$ ideals contains a subrng then one of them must contain the subrng [duplicate]

Let $B_1,B_2, \ldots , B_n$ be ideals in $R$ and at least $n-2$ of them are prime. If $S \subset R$ is a subrng (without identity) contained in $B_1 \cup \ldots \cup B_n$ then one of $B_i$'s must ...
2
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0answers
37 views

Euler's Totient Function in Other Rings

I'm looking for rings other than the integers on which we could define an interesting analogue of Euler's Totient function. E.g., on a Euclidean domain with norm $N$ we could let $\phi(x) = ...
1
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1answer
20 views

Relation of characteristic of a field and a domain.

Let $A,B$ be two integral domains and $\varphi : A \longrightarrow B$ a ring homomorphism. Assume that $A$ is a field, how are the characteristics of $A$ and $B$ related? What I found so far: ...
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1answer
41 views

Given a ring $R$, $R=(1)$ is a principal ideal

An ideal generated by the element $a$ is defined to be the intersection of all ideals containing $a$. My book says $R=(1)$ is a principal ideal, and I know how to convince myself of this using ...
0
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1answer
25 views

Fractional ideals in the quotient field of Dedekind

Let $R$ be a Dedekind ring, $K$ its quotient field. If $J$ is a fractional $R$-ideal in $K$ then I want to show that $KJ=K$, so that it's a full $R$-lattice in $K$. Since $J$ is non-zero, we can ...
2
votes
2answers
56 views

Correspondence between nilpotents and between idempotents

It is well-known and easily proved that whenever $R$ is a commutative ring with unity and $S$ is a multiplicative subset of $R$, each ideal of the localization ring $R_S$ is an extended ideal (with ...
0
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1answer
42 views

Examples and counter-examples for rings

Here's what I am trying to do: Listing mnemonics used: $D_n(\Bbb R)$ to denote diagonal $n \times n$ matrices with real coefficients. I.D. - Integral domain, E.D. Euclidean Domain Much of the ...
2
votes
1answer
69 views

Prove that $m_1m_2\ldots m_r=n_1n_2\ldots n_s$ implies $r=s$ for distinct maximal ideals

Let $R$ be a commutative ring where $m_1,m_2,\ldots,m_r$ and $n_1,n_2,\ldots,n_s$ are maximal ideals such that $m_1m_2\ldots m_r=n_1n_2\ldots n_s$ and $m_i \neq m_j$, $n_i \neq n_j$ if $i \neq j$. ...
0
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1answer
35 views

Proving $R/J$ is local, where $R=k[\Gamma]$ and $J=(x^1)\unlhd R$.

Let $\Gamma$ be the set of symbols of the type $x^q$, where $q\in\Bbb Q, \;q\ge0$. Setting $x^{q_1}\cdot x^{q_2}:=x^{q_1+q_2}$, $(\Gamma,\cdot)$ becomes a semigroup. Let then $k$ be a field. Let's ...
1
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1answer
27 views

Number of Associated Prime Ideals vs. Number of Maximal Ideals

This is a very naive (and presumably basic) question, but suppose you have an ideal $I \subset R$ in a commutative Noetherian ring with unity. Does the number of maximal ideals containing $I$ have ...
4
votes
2answers
90 views

Count the number of elements of ring [closed]

1/ How to count the number of elements of $\mathbb{Z}[i]/(1+2i)^n$? 2/ How to write $\mathbb{Z}[i]/(1+2i)^n$ as direct sum of cyclic groups (in view of the structure theorem of finite abelian ...
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3answers
35 views

In the proof of the universal property of tensor product of modules in Dummit and Foote

The following is Theorem 8 in Chapter 10.4 (p.362 3rd edition) of Abstract Algebra by Dummit and Foote: Let $R$ be a subring of $S$, let $N$ be a left $R$-module and let $\iota:N\to S\otimes_R N$ ...
1
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1answer
22 views

Is being in the same ideal class transitive?

Two nonzero ideals $\mathfrak{a}$ and $\mathfrak{b}$ in a Dedekind domain are in the same ideal class if there exist nonzero elements $x$ and $y$ such that $x \mathfrak{a} = y \mathfrak{b}$. My ...
0
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0answers
50 views

Is it true that taking injective hull commutes with the tensor product?

Let $M$ and $N$ be two modules (can assume them to be finitely generated if need be) over the ring $A=k[x_0,...,x_n]$. Denote by $E(M)$ the injective hull of $M$. We work in the category of positively ...
3
votes
1answer
48 views

Are the quaternions a domain?

I have to give an example of a non-commutative domain that is not a division ring. My first thought was $R = \big\{ a + bi + cj + dk \mid a,b,c,d \in \mathbb{Z} \big\}$ since $R$ is clearly ...