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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0
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1answer
26 views

Why the map sending $a\otimes b$ to $(ab, a\bar{b})$ is injective?

I am wondering why the ring homomorphism $\phi : \mathbb{C}\otimes_\mathbb{R}\mathbb{C} \longrightarrow \mathbb{C}\times\mathbb{C}$ sending $a\otimes b$ to $(ab, a\bar{b})$ is one-to-one. So any ...
1
vote
1answer
95 views

$\mathbb C[x_1,\ldots,x_n]/I=\mathbb C\times\cdots\times\mathbb C$.

Let $A=\mathbb C[x_1,\ldots,x_n]/I$ and for every $y\neq 0$, we have $y^2\neq 0$ and $\dim A=0$. I would like to prove that $A=\mathbb C\times\cdots\times\mathbb C$. Attempt of a solution $\dim A=0$...
0
votes
1answer
59 views

Intersection of $max(R)$ with a closed subset in $Spec(R)$

Let $R$ be a commutative ring with unity and $E$ be a nonvoid closed subset of $Spec(R)$. If $U$ is an open subset of $Spec(R)$ with $E∩Max(R)⊆U$, where $Max(R)$ is the set of maximal ideals of $R$, ...
2
votes
1answer
297 views

Jacobson radical of a ring finitely generated over $\mathbb Z$

If a commutative ring $R$ with $1$ is finitely generated over $\mathbb Z$ could one deduce that the Jacobson radical of $R$ is nilpotent? I am aware of the well-known fact that when $R$ is artinian,...
5
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0answers
58 views

Is there an online database somewhere that lists identities for algebraic structures with two binary operators?

I'm working on an abstract algebra library in Python, and I'm trying to include as many functions that analyze algebraic structures, returning true or false based on whether or not the algebra ...
0
votes
1answer
27 views

Proving that a zero ring cannot have a homomorphism to a unital ring.

Q: Prove that a zero ring cannot have a homomorphism to a unital ring. I don't know how to prove this. If $f:R\to S$ is the homomorphism, and $f(0_R)=0_S$, then I don't see which homomorphism ...
2
votes
0answers
41 views

An inverse limit of a certain inverse system

Let $∆$ be a directed set and $(N_i,f_{ji})_{i∈∆}$ be an inverse system of $R$-modules. Fix $α \in∆$ and consider $(M_i,g_{ji})_{i\in∆}$ as follows: $M_i=N_i$ for $i≥α$, $M_i=0$ for $i<α$, and $...
0
votes
1answer
42 views

Prove that if $\mathrm D$ is integral domain, then $\mathrm D$is UFD iff these conditions hold.

Prove that id $\mathrm D$ is integral domain, the "$\mathrm D$ is UFD " iff (1) $\mathrm D$ satisfies the ACC for principal ideals. (2) every irreducible element is a prime element. I've ...
6
votes
2answers
56 views

Is $(x^2 + 1, y^2 + 1)$ a prime ideal in $\mathbb{Q}[x, y]$?

At first I was looking for a ring homomorphism from $\mathbb{Q}[x, y]$ to a domain with $(x^2 + 1, y^2 + 1)$ as it's kernel, but I could not find one. Now I am thinking: maybe $(x + y)(x - y) = x^2 ...
4
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3answers
137 views

Why $\mathbb{C}\otimes_\mathbb{Q}\mathbb{C} \not\cong \mathbb{C}\times\mathbb{C}$

I am wondering if this is true or not: $\mathbb{C}\otimes_\mathbb{Q}\mathbb{C} \cong \mathbb{C}\times\mathbb{C}$ (as rings). Any help or suggestion would be helpful.
1
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2answers
270 views

Number of non trivial ring homomorphisms from $\mathbb Z_{12}$ to $\mathbb Z_{28}$ [duplicate]

One homomorphism is $ 1 \mapsto 1$. Other homomorphisms are: We know that if $f$ is a homomorphism from $R$ to $S$ and $f(1_R) \neq 1_S$, then $f(1_R)$ is a zero diviasor in $S$. So zero divisor ...
0
votes
3answers
74 views

Why is the quotient ring $\frac{\mathbb{R}}{\mathbb{R}} = \left \{ 0 \right \}$?

As the question says, why is the quotient ring $\frac{\mathbb{R}}{\mathbb{R}} = \left \{ 0 \right \}$ ? Shouldn't it be $\mathbb{R}$? We have that $\frac{\mathbb{R}}{\mathbb{R}} = \left \{ x + \mathbb{...
0
votes
1answer
105 views

Prove an ideal is maximal

Question Prove the ideal $\mathrm I=\{f \in \mathrm R| f(2)=0 \}$ of $\mathrm R=\{f(x) | f: \Bbb R \to \Bbb R $ is continue} is maximal. DO NOT use the $1$st isomorphism theorem. I ...
3
votes
0answers
88 views

Finding the prime element of a place of a function field of a elliptic curve.

Let $p$ be an elliptic curve in $\mathbb{C}[X,Y] $. Consider the quotient ring $A = \mathbb{C}[X,Y]/(p) $ and its field of fractions $F = frac(A) $. For all $f + (p) \in A$, define $deg_A(f + (p))= ...
0
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0answers
42 views

Complexity of and an algorithm for finding ideals of a ring?

One of the problems that has been a roadblock in my understanding of ideals has been how one would find them. One way of finding an I of some ring R would be to say $ \forall x \in I, \forall r \in R ...
1
vote
1answer
146 views

Proving a ring is Euclidean domain

Show that $\mathbb Z[\sqrt 2i]=\{a+bi\sqrt2|a,b \in \mathbb Z\}$, with a function $V: \mathbb Z[\sqrt 2i]/\{0\} \rightarrow \mathbb N$ defined $V(a+bi\sqrt2)=|a^2+2b^2|$ is a Euclidian domain. I ...
0
votes
2answers
52 views

Polynom irreducibility over $\mathbb Z_5$

So, I have polynom over $\mathbb Z_5$: $x^8 - x^7 + 2x^6 + x^5 + 2x^4 + 2x^2 +3x +1$ and I have to find his irreducibile factors. How to do that? I can find his roots by replacing $x$ with $[1]_5, [2]...
2
votes
1answer
58 views

ideal calculations: $2\mathcal{O}_K=\mathfrak{B}^4$ in the ring of integers of $K=\mathbb{Q}(i,\sqrt{2m})$

Let $K=\mathbb{Q}(i,\sqrt{2m})$ where $m \in \mathbb{Z}$ is odd and squarefree. Let $\alpha = (1+i)\sqrt{2m}/2$. Then $\alpha^2=im$, such that $\alpha$ is part of the ring of integers $\mathcal{O}_K$. ...
1
vote
2answers
65 views

Question about kth root of a reduced ring element.

Let $n > 1$ be a positive integer. Let $k > 1$ be a positive integer. Define the reduced polynomial rings $f_n = \Bbb R[X_n]/(1+(X_n)^{n})$ How do we know if $(X_n)^{1/k}$ is an element of $...
4
votes
1answer
200 views

Integral domain with a finitely generated non-zero injective module is a field

Suppose that $R$ is a integral domain. Suppose that there exists a non-zero finitely generated injective module $M$. How can I prove that $R$ is field?
2
votes
1answer
69 views

Quotients of $p$-adic completion

Let $R$ be a commutative ring and $p \in R$. Consider the $p$-adic completion $\widehat{R} := \varprojlim_{n} \, R/p^n$. When do we have $\widehat{R}/p^n \widehat{R} \cong R/p^n R$? For fixed $n$ ...
0
votes
0answers
224 views

Generalization of Chinese Remainder Theorem to infinite ideals

I'm looking for any (obviously weaker) generalization of this famous theorem in the special case that the family of ideals is not finite.
1
vote
1answer
67 views

Number of solutions of some congruence equations.

How many $[u]\in(\mathbf{Z}/ab\mathbf{Z})^\ast$ satisfy the equations $u\equiv 1 \bmod \ a$, $u\equiv 1 \bmod \ b$? I somehow believe that the answer might be $(a,b)$. Is this actually true? Is the ...
2
votes
1answer
93 views

Noetherian rings/Hilbert's Basis Theorem [closed]

So I'm studying the proof of Hilbert's Basis Theorem - we've shown that $λ(I)$ is an ideal of $R$ and and then it says "Since $R$ is Noetherian, we have $λ(I) = \sum\limits_{i=1}^k s_iR$ for some $s_1,...
0
votes
1answer
25 views

Counter-example to show that difference doesn't hold in ideal

Let $I=\left \langle a \right \rangle$ and $J = \left \langle b \right \rangle$ be ideals in a ring A. Show that $I\cdot J = \{xy\mid x\in I,y\in J\} $ is an ideal in A and $I\cdot J = \left \langle ...
0
votes
3answers
258 views

Primitive Root in Quotient Ring

Find a primitive root of $R[x]/\langle x^4+x+2 \rangle$ where $R$ is the integers mod $3$. Is there a good general stratagy to this sort of thing?
1
vote
1answer
188 views

$2$-dimensional Noetherian integrally closed domains are Cohen-Macaulay

Any 1-dimensional Noetherian domain is Cohen-Macaulay (C-M). For the $2$-dimensional case, a condition of being integrally closed is necessary to be added for a Noetherian domain to be C-M, which ...
5
votes
2answers
846 views

Showing that minimal polynomial has the same irreducible factors as characteristic polynomial

I'm trying to show that the minimal polynomial of a linear transformation $T:V \to V$ over some field $k$ has the same irreducible factors as the characteristic polynomial of $T$. So if $m = {f_1}^{...
1
vote
2answers
86 views

$A = \bigcap_{\mathfrak{p} \in \text{Spec(A)}} A_{\mathfrak{p}} = \bigcap_{\mathfrak{m} \in \text{MaxSpec(A)}} A_{\mathfrak{m}}$

I'm doing this exercise. Let $A$ be an integral domain, then prove that $$A = \bigcap_{\mathfrak{p} \in \text{Spec(A)}} A_{\mathfrak{p}} = \bigcap_{\mathfrak{m} \in \text{MaxSpec(A)}} A_{\mathfrak{...
1
vote
1answer
107 views

Free modules over local Artin rings

Let $A$ be a local Artin ring, $B$ be a $A$-flat algebra and $M$ a finitely generated $B$-module and flat $A$-module. Let $k$ be the residue field of $A$ i.e., $A/m$ where $m$ is the maximal ideal of $...
2
votes
2answers
57 views

What the rings between Principal Ideal Ring and Field of fractions are?

Prove that all of the rings, which mediate between principal ideal ring $K$ and the field of fractions $Q$, are the principal ideal ring.
2
votes
1answer
121 views

A question on localization of fractional ideals

I have a domain $A$ with field of fractions $K$ and a non-zero fractional $A$-ideal $I$. Let $I^{-1}$ be the fractional ideal $\{a\in K\mid aI\subseteq A\}$. I assume that $II^{-1}\subseteq \...
2
votes
1answer
84 views

Properties of the element $2 \otimes_{R} x - x \otimes_{R} 2$

I'm doing this exercise from Dummit-Foote: Let $I = (2, x)$ be the ideal generated by $2$ and $x$ in the ring $R = \mathbb{Z}[x]$. Show that the nonzero element $2 \otimes_{R} x - x \otimes_{R} ...
4
votes
1answer
74 views

Eisenstein integers and $\mathbb{Z}C_3$

The Eisenstein Integers $a+b\omega$ with norm $N(x)=a^2-ab+b^2$ form a commutative ring, as does the group ring $\mathbb{Z}C_3=\{\sum_{g\in C_3} a_g g \mid g \in C_3, a_g \in \mathbb{Z}\}$. $\mathbb{Z}...
2
votes
1answer
162 views

Number of ring homomorphisms from a finite field to a ring.

How many homomorphisms are there from a finite field to a ring? I have some basic knowledge of this but I am not able to put it into use. I know $1)$ If $\phi$$(1)=a$, then $|a|$ should divide both, ...
1
vote
2answers
371 views

Grade of an ideal in a Noetherian ring

I want to prove that if $R$ is a Noetherian integral domain and $I$ is a nonzero ideal of $R$, then $I^{-1}=R$ if and only if $\operatorname{grade}(I)≥2$. For the "only if" part, I say $II^{-1}=R$ ...
4
votes
1answer
143 views

Showing that $x^n -2$ is irreducible in $\mathbb{Q}[X]$

I'm trying to show that the polynomial $X^n -2$ ($n \in \mathbb{N}$) is irreducible in $\mathbb{Q}[X]$ but am a bit stuck. Methods I know to show irreducibility: Gauss' lemma - which says that if I ...
1
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0answers
60 views

Does This Ring have a Name?

Let $M_1=\{0,1,2,4,5,8,9,10,\cdots\}$ be the set of nonnegative integers that can be written as a sum of two perfect squares. Let $M_2=\{\sqrt{m}: m\in M_1\}=\{0,1,\sqrt{2},2,\sqrt{5},\cdots\}$. Let $...
3
votes
2answers
30 views

Find zero and one element for given ring

Given $ M$ and the powerset $\mathcal{P}(M) $, let $A,B \subseteq M$ and define: $A \bigoplus B := A \backslash B \cup B \backslash A$ $A * B := A \cap B$ Then is $R_M = \langle \mathcal{P}(M); \...
2
votes
1answer
55 views

$\mathbb Q$ Field extension

Consider the Field $F = \mathbb Q(2^{\frac 1 3})$, Is $\sqrt 2 \in F$? I'm trying to figure out how to determine that and similar questions, can you give me a hint or some guidance on how to do that?
4
votes
1answer
97 views

Direct product of Cohen-Macaulay rings/Eisenbud, Exercise 18.6

Somehow I believe (or doubt (!)) that direct product of two Cohen-Macaulay (C-M) rings may not be C-M. Can anybody give me an example verifying this? I would be grateful to him/her.
4
votes
2answers
523 views

Inverting $a+b\sqrt{2}$ in the field $\Bbb Q(\sqrt{2})$

I have been reading through my notes and I came across this example and I found it hard to understand so I need some help in explaining how the inverse of this is found. The set $\mathbb Q(\sqrt 2)=\{...
2
votes
1answer
65 views

Prove that $B_{q}$ is flat over $B_{p}$

I'm doing this exercise in "Introduction to Commutative Algebra" of Atiyah and get confused by the hint in this book. Here is the exercise: Let $f: A \rightarrow B$ be a flat homomorphism of ...
3
votes
0answers
579 views

Tensor product of the spaces of quaternions and complex numbers

Let $ \mathbb{H} $ be the ring of quaternions and make the vector space $A = \mathbb{H} \otimes \mathbb{C}$ into a ring by defining $$(a \otimes w)(b \otimes z) = (ab \otimes wz) $$ for $a,b \in \...
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vote
0answers
34 views

Showing isomorphism between quotient rings

Let $f: R_1 \rightarrow R_2$ be a surjective ring homomorphism. Let $I_1$ be defined: $I_1=\{r_1\in R_1| f(r_1)\in I_2\}$, and I've showed that $I_1 \lhd R_1$. It's also known that $I_2 \lhd R_2$. ...
2
votes
1answer
55 views

understanding roots of polynomials in field extensions

I'm running into a conceptual stumbling block understanding the application of the FHT to field extensions and finding roots, if anyone has any pointers on where I might be misunderstanding. I'm ...
4
votes
3answers
58 views

Let $f: R_1 \rightarrow R_2$ be a ring homomorphism. Does $r \in R_1$ being non-invertible mean $f(r)$ is non-invertible?

Let $f: R_1 \rightarrow R_2$ be a ring homomorphism. Does $r \in R_1$ being non-invertible mean $f(r)$ is non-invertible? This seems incorrect, but I find it difficult finding counter-example... ...
4
votes
1answer
100 views

Do $IJ$ and $I\cap J$ coincide if $I$ and $J$ are coprime? Also if ring $R$ has a $1$ and is not commutative?

Let $R$ be a ring (with identity) and let $I,J$ be two coprime (two-sided) ideals in it. In Algebra: Chapter $0$, Aluffi, III. exercise 4.5. the reader is asked to prove that: $$IJ=I\cap J$$ ...
1
vote
0answers
117 views

Sufficient conditions for quotient ring to be Cohen-Macaulay

We know that every Noetherian integral domain with (Krull) dimension $1$ is Cohen-Macaulay (CM). In a commutative algebra text the author have presented the following problem: "Let $(R,m)$ be a CM ...
3
votes
2answers
124 views

All homomorphisms from a simple ring to a non-zero ring are injective

Let $R$ be a simple ring and $T$ be a non-zero ring. Let $f\colon R \rightarrow T$ be a ring homomorphism. Show $f$ is injective. Proof: $\ker f \lhd R$, so $\ker f=R$ or $\ker f=\{0\}$. If $\ker f=R$...