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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2answers
34 views

State a reason the given function is not a homomorphism

$f:\Bbb R \rightarrow \Bbb R$ and $f(x)=\sqrt x$ For $\forall x\lt0\in\Bbb R$, $f(x)=\sqrt x\in\Bbb C\notin\Bbb R$ Does my answer make sense, or should I elaborate with words?
23
votes
1answer
453 views

Subring of $\mathcal O(\mathbb C)$

Let $\mathfrak A \subset \mathcal O(\mathbb C)$ be the subring generated by the nowhere zero analytic functions $f: \mathbb C \to \mathbb C$. Do we have a precise description of $\mathfrak A$? Is ...
12
votes
1answer
251 views

$y^2 = x^3 - 26$, exist ideal satisfying conditions?

For the solution $(x, y) = (3, 1)$ of $y^2 = x^3 - 26$, does there necessarily exist an ideal $I$ of the integer ring $\mathbb{Z}[\sqrt{-26}]$ of $\mathbb{Q}(\sqrt{-26})$ such that $(y + ...
2
votes
0answers
40 views

Application of the structure theorem for finitely generated modules over a PID

I've been trying to figure out the correct way to do this and I'm not sure I've got it quite right. $\mathbb{Z}^4/S$ where $S = \{(4, 12,20,8),(6,30,18,12),(4,16,16,8),(8,40,24,16) \}$ So, $M_0 = ...
0
votes
1answer
35 views

Non-zero maps between modules

Let $R$ be a ring and $M,N$ two $R$-modules such that there exists a non-zero map $\psi : M \to N$. Is it true that there exists a non-zero map $\varphi: N \to M$ ? I do not have any particular ...
0
votes
1answer
47 views

Element invertible in integral extension of ring implies invertible in ring [duplicate]

Please excuse some minor hiccups in terminology, I am primarily reading this in Swedish so feel free to correct any. Let $A\subseteq B$ be an integral extension and $\alpha\in A$ an invertible ...
0
votes
1answer
39 views

Ring Structure for Non Commutative Groups: Is there a grander reason for Abelian requirements?

So I was considering the following idea. Let a generalized Ring $gR$ be a set equipped with two operations $$u_1, u_2$$ Such that $gR$ is a with the operation $u_1$ and the operation $u_2$ ...
1
vote
1answer
18 views

Every modular right ideal is contained in a modular maximal ideal

If $R$ is a ring, possibly without $1$, a right ideal $\mathfrak{a}$ of $R$ is modular if there exists $e\in R$ such that $r-er\in \mathfrak{a}$ for all $r\in R$. So $e$ is a left identity mod ...
0
votes
1answer
39 views

$\mathbb{Z}$ is Euclidean domain

It is well known that $\mathbb{Z}$ is Euclidean domain. But when my teacher asked me to prove it, I went towards Peano Axioms for natural numbers, or integers. Question: How can we prove that ...
1
vote
1answer
52 views

Can we always write $gcd(x,y)$ as $ax+by$ in UFD?

Let $R$ be a commutative ring with unity. Now assume that $R$ is Unique Factorization Domain, but not necessarily Principal Ideal Domain. Question: Let $x,y\in R$ be such that their GCD exists in ...
6
votes
1answer
138 views

What exactly is Hensel doing for us in this result?

I'm reading a paper where the author appeals to Hensel's lemma, but it is not clear to me quite how it is meant to be applied (or, for that matter, which version!). My commutative algebra background ...
3
votes
1answer
49 views

converging power series over $p$-adic integers is a UFD

Denote by $|\cdot |$ the $p$-adic norm, and let $$\mathbb Z_p \{z\}=\left\{\sum_{n=0}^\infty a_nz^n;\ a_n\in \mathbb Z_p ,\ |a_n|\underset {n\rightarrow \infty} {\longrightarrow 0} \right\}$$ the ...
0
votes
1answer
129 views

In what structures does $ (-1)^2 = 1$?

Does $ (-1)^2 = 1$ anywhere you have associativity and an inverse element? Thanks!
3
votes
1answer
44 views

Looking for a terminology in ring theory (“ideal” which is not necessarily closed under addition )

I am wondering if there is a name for the subsets $S$ of a commutative ring $R$ such that for every $r\in R$ and every $s\in S$ we have $rs\in S$. Thus $S$ is ...
3
votes
1answer
46 views

Is a = 0 a valid counterexample to this statement?

This is an exercise in a text I am reading for a ring theory course. Suppose the ring R contains element a such that 1) a is idempotent and 2) a is not a zero divisor of R. Deduce that a serves as a ...
4
votes
1answer
78 views

Under what conditions does $M \oplus A \cong M \oplus B$ imply $A \cong B$?

This question is fairly general (I'm actually interested in a more specific setting, which I'll mention later), and I've found similar questions/answers on here but they don't seem to answer the ...
1
vote
1answer
52 views

What is the difference between submodules of $A/\mathfrak a$ as an $A$-module or as an $A/\mathfrak a$-module?

If $\mathfrak a$ is an ideal of unital commutative ring $A$, Then we can see to $A/\mathfrak a$ as an $A$ module or as an $A/\mathfrak a$ module. If $A=\mathbb Z$ there is no structure difference ...
1
vote
1answer
49 views

Show that every short exact sequence $0\rightarrow M' \rightarrow M \rightarrow M'' \rightarrow 0$ (with $M''$ free) is split exact

Note: $M,M',M''$ are modules. In order to show that it's split exact, I understand that I need to show that $\beta:M\rightarrow M''$ has a left inverse, and similarly that $\alpha:M'\rightarrow M$ ...
0
votes
0answers
23 views

recover (pontrjagin) ring structure from the localization (w.r.t. $\pi_0$)

Let $R$ be a ring and $S$ a given multiplicative subset of $R$. Suppose we know the multiplication structure of $S$. If we know the ring structure of $R[S^{-1}]$, the localization of $R$ with respect ...
2
votes
0answers
24 views

Mixed vs Equal Characteristic Local Rings

This is more of a vague, intuitive type of question, so perhaps there isn't anything too concrete anyone can offer. I am trying to get a sense of precisely why working with local rings of equal ...
3
votes
3answers
467 views

Example of a ring with infinitely many zero divisors and finitely many invertible elements

I am preparing to my abstract algebra exam and I try to find an example of a ring with infinitely many zero divisors and finitely many invertible elements (rather simple if possible). Does it even ...
2
votes
1answer
30 views

Example of $Q((x))$ that doesnt match field of fractions of ring $F[[x]]$

Let $F$ be a commutative ring without zero divisors and $Q$ -its field of fractions. Let $Q(x)$ be also field of fractions of ring $F[x]$. How can field $Q((x))$ not match field of fractions of ring ...
25
votes
5answers
1k views

Why is the commutator defined differently for groups and rings?

The commutator of two elements in a group is defined as $[g, h] = g^{−1}h^{−1}gh.$ In a ring, the commutator of two elements is $[a, b] = ab - ba.$ I'm asking because a ring is a (abelian) group ...
0
votes
1answer
44 views

Is there a way to generate groups, rings, fields, etc.? [closed]

There are ways to generate a list of numbers $a_1, a_2,...a,n$ such that no $a_i,a_j$ share any factors, mainly by letting $a_{n+1}=a_n^2-a_n+1$ with $a_0>1$, my question is: is there a way to ...
3
votes
1answer
118 views

What motivates the definition of a ring in abstract algebra? [closed]

I've read a lot of questions about rings but I still don't know why they are useful/ I'm sure they are, but I don't know why. Are their properties somehow used in proofs or as foundations for ...
2
votes
2answers
34 views

There is no nontrivial ring homomorphism between two commutative rings with unity and characteristic of distinct primes

The following is an old exam question and the question is: Show that there is no nontrivial ring homomorphism between two commutative rings with identity if their characteristics are distinct primes. ...
8
votes
1answer
81 views

$A$ regular, $k'/k$ transcendental. How to prove that $A \otimes_k k'$ is regular?

Let $k$ be a field and $k'$ a purely transcendental extension of $k$. Let now $A$ be an integral finitely generated $k$-algebra. How to prove that if $A$ is regular then $A \otimes_k k'$ is also ...
3
votes
2answers
86 views

The subring of $k[x, y]$ generated by $\{x y^{i}: i\geq 0\}$ is not finitely generated over $k$ [duplicate]

Let $R$ be the subring of $k[x, y]$ generated by $\{x y^{i}: i\geq 0\}$. Can someone explain why $R$ is not finitely generated as a ring over $k$ (i.e. finitely generated as a $k$-algebra)? By ...
1
vote
2answers
126 views

Prime ideals in $R[x]$, $R$ a PID

Let $R$ be a PID. Show that if $\mathfrak p \subset R[x]$ is a prime ideal, $(r) = \left\{h(0) \colon h(x) \in \mathfrak p \right\}$, and $$\mathfrak p = (r, f(x), g(x)),$$ where $f(x), g(x) \in ...
5
votes
2answers
82 views

Prime ideal $P$ of $\mathbb{Z}[x]$ such that $P \cap \mathbb{Z}=\{0\}$ is principal

The problem stated more precisely is this: Let $P$ be a prime ideal of $\mathbb{Z}[x]$ such that $P \cap \mathbb{Z} =\{0\}$. Show that $P$ is a principal ideal. I think there is a problem with my ...
3
votes
1answer
33 views

The quotient of a direct sum of rings

Is the quotient of a direct sum of rings isomorphic to the direct sum of the quotients? $$ (R_1 \oplus R_2 ) / \langle (x_1, x_2)\rangle = (R_1 / \langle x_1\rangle ) \oplus (R_2 /\langle x_2\rangle) ...
1
vote
2answers
48 views

The ideal generated by the set $I \cup \{a\}$.

Let $I$ be an ideal of $R$ a commutative ring with identity. For some $a \in R$, prove that the ideal generated by $I \cup \{a\}$, denoted $(I,a) = \{ i + ra \mid i \in I \text{ and } r \in R\}$. My ...
1
vote
1answer
36 views

Example of a Non-Graded Ideal in a Graded Ring

A ring $S$ is said to be graded if there are additive subgroups $S_0, S_1, S_2, \ldots$ such that $S=\bigoplus_{k\geq 0}S_k$ and $S_iS_j\subseteq S_{i+j}$ for all $i$ and $j$. An ideal $I$ in a ...
3
votes
4answers
57 views

If $R$ is a ring with identity and $a$ is a unit, prove that the equation $ax=b$ has a unique solution in $R$.

So, this was my initial proof: Assume $R$ is a ring, and $a,b\in R$ Let $x_1$ and $x_2$ be solutions of $ax=b$ Hence, $ax_1=b=ax_2 \Rightarrow ax_1-ax_2=0_R \Rightarrow a(x_1-x_2)=0_R$ Thus, we ...
7
votes
2answers
166 views

Expand $\binom{xy}{n}$ in terms of $\binom{x}{k}$'s and $\binom{y}{k}$'s

Motivated by this question, I want to find a complete set of relations for the ring of integer-valued polynomials, where the generators are the polynomials $\binom{x}{n}$ for $n\in \mathbb{N}$. The ...
0
votes
1answer
21 views

What is a relation (finitely related module)?

https://en.wikipedia.org/wiki/Finitely-generated_module#Finitely_presented.2C_finitely_related.2C_and_coherent_modules I've understood the first part of the definition. Then, "M is isomorphic to ...
2
votes
2answers
58 views

Can one decompose any ring into a (possibly infinite) product of indecomposable rings?

Let $R$ be a ring. Do there exist indecomposable rings $R_i$, $i\in I$, such that $R={\prod}_{i\in I} R_i$?
0
votes
1answer
20 views

I is equal to the preimage of its image.

Lemma. Let $f$ be a homomorphism from the ring $R$ onto the ring $R'$. If $I$ is any ideal of $R$ such that $\ker(f)$ is a subset of $I$, then $I = f^{-1}(f(I))$. I am trying to understand this ...
3
votes
1answer
61 views

Isomorphic or equal?

Let $\sim_n$ be the usual equivalence relation of congruence modulo $n$ in $\mathbb{Z}$, i.e., for $a,b\in\mathbb{Z}$, $a\sim_nb\Leftrightarrow a-b=k\cdot n$ for some $k\in\mathbb{Z}$. For $n=0$ the ...
0
votes
4answers
53 views

If $I$ and $J$ are distinct ideals in ring $R$ and $f:R \to R'$ is a homomorphism then is $f(I) = f(J)$?

The text book I am reading says that if $I$ is a subset of $J$ and $J$ is a subset of $I + \ker (f)$ then $f(I) = f(J)$. The argument goes: $f(I)$ is a subset of $f(J)$ is a subset of $f(I + \ker (f)) ...
0
votes
0answers
30 views

Proof that the kernel is a normal subgroup of the domain: repeated line

On proofwiki (https://proofwiki.org/wiki/Kernel_is_Normal_Subgroup_of_Domain), the lines corresponding to 'definition of identity' and 'definition of kernel' are identical. Why do we need the second ...
2
votes
1answer
27 views

is $\mathbb{Q} ^{|\mathbb{R}|} ‎\times‎ \{1 \}$ divisible subgroup of $ \mathbb{Q} ^{|\mathbb{R}|} ‎\times‎ \mathbb{Z}_2$?

According to Unit Groups of Classical Rings by Karpilovsky, p.107 we know that: If $F$ is a real-closed field, then $F^*‎\simeq‎ \mathbb{Q} ^{|F|} ‎\times‎ \mathbb{Z}_2$. Now, we know that ...
3
votes
1answer
46 views

A ring isomorphic to its square [duplicate]

Is there an example of a ring $A$ (with unity) which is isomorphic as unital rings to $A\times A$? Any such ring can't have invariant basis number so in particular can't be commutative.
1
vote
1answer
59 views

Prove that $f$ is a nonzerodivisor on $R[x_1,\dots,x_r]/IR[x_1,\dots,x_r]$ for every ideal $I$ in $R$

Let $R$ be a Noetherian commutative ring with unity, and $S=R[x_1,\dots,x_r]$. Let $f\in S$ be a nonzerodivisor of $S$. Suppose that the ideal generated by the coefficients of $f$ is $R$. How to ...
2
votes
2answers
43 views

Show that the ideal $I=\left\langle x_1^2+1,x_2,…,x_n\right\rangle$ is maximal in $\mathbb{R}[x_1,…,x_n]$.

This is an exercise in "Ideals, varieties, and algorithms" by Cox et al. It first asks to show that $I=\left\langle x^2+1\right\rangle$ is maximal in $\mathbb{R}[x]$. I can show it because it is a ...
0
votes
2answers
33 views

Find character table for symmetric group $S_3$

This group contains all permutations of 3 elements, so it has order 3!=6. Its three congruency classes are {1}, {(1,2),(1,3),(2,3)}, {(123),(132)}. As we know that the number of congruency classes ...
1
vote
4answers
52 views

$\mathbb Z_n$ is semisimple iff $n$ is square free

$\mathbb Z_n$ is J-semisimple iff $n$ is square free. A ring $R$ is said to be $J$ semisimple if intersection of all maximal ideals of $R$ is $\{0\}$. If $n$ is square free then ...
1
vote
1answer
70 views

Structure of the unit group $(\mathbb{Z}[i]/8\mathbb{Z}[i])^\times$

I know that $\mathbb{Z}[i]/8\mathbb{Z}[i]=\{a+ib \mid a,b\in\mathbb{Z}_8\}$. But I'm not able to comprehend what $(\mathbb{Z}[i]/8\mathbb{Z}[i])^\times$ is. Can someone please help me get its ...
3
votes
0answers
24 views

Why is this done? (Quadratic integer rings definition) [duplicate]

From Dummit & Foote pg. 229: Let $D$ be a squarefree integer. It is immediate from the addition and multiplication that the subset $\Bbb{Z}[\sqrt{D}] = \{a + b \sqrt{D} | a,b, \in \Bbb{Z}\}$ ...
1
vote
1answer
33 views

Quasi Injective vs Pseudo Injective modules

$\textbf{Definition:}$ A left $R$ module $M$ is called QI (PI) - module if for every submodule $N$, any $R$-homomomorphism (monomorphism) $N\rightarrow M$ extends to an endomorphism of $M$. For QI - ...