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
20 views

What is a prime ideal?

I am having some trouble understanding the concept of a prime ideal in ring theory. I have researched what a prime ideal is and the simplest answer I got was this: An ideal $P$ of a commutative ...
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0answers
104 views
+50

$y^2 = x^3 - 26$, 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 + ...
3
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1answer
37 views

Litterature on noncommutative ring

I am looking for books or notes about non commutative rings with with a maximum of data exposed without the help of modules (because I have many references which deal with the subject but modules are ...
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2answers
37 views

Domain and primes ideals [on hold]

Let $A$ be a commutative ring with identity and $A[x]$ a polynomial ring. Show that the ideal $(x)$ is prime in $A[x]$ iff $A$ is domain.
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1answer
40 views

performing a power operation ($a^n$) in a ring

In a ring - when performing a power operation, i.e $a^n$, to which operation is it related to? $+$ or $*$? On one hand - I know that a power is defined on multiplication - in "regular" numbers, but ...
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1answer
48 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 ...
3
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0answers
23 views

Question concerning a minimal faithful left ideal in an artin algebra

Let $B$ be an artin algebra an suppose there is a faithful projective-injective left $B$-module. Moreover, there is a minimal faithful left ideal $Be$ for some idempotent $e\in B$. 1) What does ...
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0answers
182 views
+50

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

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

The quotient of a ring by the annihilator of an ideal

Let $R$ be a commutative ring with identity and $I$ an ideal of $R$. It's true that we have an $R$-module isomorphism $$I\cong R/ann_RI,$$ where $ann_RI=\{x\in R:xr=0,\;for\;all\;r\in I\}$ is the ...
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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?
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2answers
101 views

Why doesn't $xa = x$ for all $x \in R$ imply that $a$ is the unit of $R$?

We have a ring $R$ as follows: Why is it not enough to conclude that $a$ is a unity if $xa = x$ for all $x$ in $R$? Is it because it is by definition that the unity satisfies $ax = xa = x$ for all ...
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1answer
48 views

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

Let $R$ be a PID. Show that if $r \in R$ and $$p = (r, \underline{f}(x), \underline{g}(x))$$ is prime, where $\underline{f}(x), \underline{g}(x) \in R[x]$ are nonconstant irreducible polynomials, ...
2
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1answer
53 views

Example of non-commutative ring with exactly 2014 two sided-proper ideals.

Find a non-commutative ring with exactly 2014 two sided-proper ideals. Find a ring with exactly 2014 pairwise non-isomorphic irreducible modules. If it was the commutative ring i would have ...
2
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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 ...
2
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0answers
39 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
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1answer
32 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
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1answer
34 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 $u_2$ ...
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2answers
25 views

Let I and J be ideals of a ring R. Prove that the union of I and J is an ideal of R iff I is a subset of J or J is a subset of I. [duplicate]

I'd like a proof of: Let I and J be ideals of a ring R. Prove that the union of I and J is an ideal of R iff I is a subset of J or J is a subset of I. This is my report no. 3 in my subject ...
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1answer
86 views

If ring $B$ is integral over $A$, then an element of $A$ which is a unit in $B$ is also a unit in $A$.

Let $A$ be a subring of ring $B$, with $B$ integral over $A$. If $x\in A$ is a unit in $B$, then it is a unit in $A$. I know that $f(t) = t - x$ is in $A[t]$ with $f(x) = 0$, and that there ...
0
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1answer
44 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 ...
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0answers
42 views

Let I be an ideal of a ring R. Prove that the quotient ring R/I is a commutative ring if and only if ab − ba ∈ I for all a, b ∈ R. [on hold]

This is the report no. 3 of Jennylou Canlas in our subject math126 in MSU Proof: Suppose R/I is a commutative ring. Let a, b ∈ R. Then (a + I), (b + I) ∈ R/I. Since R/I is commutative , (a + I)(b ...
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1answer
103 views

Prove that $R = S_1$ or $R = S_2$

Let $R$ be a ring and $S_i$ be the subrings of $R$ such that $R = S_1 \cup S_2$. Prove that $R = S_1$ or $R = S_2$. I am not exactly sure what to do here. If I want to proceed with letting $R \neq ...
6
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1answer
133 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 ...
2
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1answer
76 views

In a commutative ring without identity, is $(a)(b)\subset (ab)$ or $(ab)\subset (a)(b)$?

Let $R$ be a commutative ring without unity. Consider an ideal $(a)$ generated by $a\in R$. Note that $(a)=\{ra+na : r\in R, n\in \textbf Z\}$ since $R$ has no identity. I wonder if $(a)(b)\subset ...
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1answer
15 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 ...
6
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2answers
280 views

Polynomial rings — Inherited properties from coefficient ring

To avoid mixing up things, I wanted to collect properties a polynomial ring is inheriting from the coefficient ring and what property implies another. Let $R$ be a ring (what else do I need at which ...
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1answer
35 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 ...
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1answer
38 views

Radical of An Ideal In $Z_n$ [closed]

I am searching for an answer in $Z_n$ regarding the radical of an ideal. Consider an ideal $(a)$ in $Z_n$. Can we calculate radical of $(a)$ in general?
0
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1answer
127 views

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

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

Polynomial-closed properties of rings [closed]

If $R$ is a ring with certain property, sometimes when we pass to the polynomial ring in one variable, the ring $R[x]$ still has the same property. For instance, it's a theorem that if $R$ is a UFD ...
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0answers
60 views

searching about an isomorphism [on hold]

I'm looking for an isomorphism : $$H: \overbrace{\mathbb{F}_q^r\oplus\cdots\oplus \mathbb{F}_q^r}^{l\text{ times}}\longrightarrow \frac{\mathbb{F}_q[X_1,\ldots,X_l]}{(X_1^r-1)\cdots(X_l^r-1)}$$
3
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1answer
42 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 ...
5
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1answer
90 views

What is the kernel of $K[x^2,x^3][T] \to K[x]$, defined by: $T \mapsto x$?

Consider $K[x^2,x^3] \subset K[x]$, where $x$ is an indeterminate over a (zero characteristic) field $K$. Clearly, $x$ vanishes the following polynomials $\in K[x^2,x^3][T]$: $f(T)=x^2T-x^3$, ...
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2answers
90 views

For which $d \in \mathbb{Z}$ is $\mathbb{Z}[\sqrt{d}]$ a unique factorization domain?

Is there a general criterion which tells me whether $\mathbb{Z}[\sqrt{d}]$, $d \in \mathbb{Z}$ is a unique factorization domain? $\mathbb{Z}[\sqrt{-5}]$ is a frequent example for non-unique ...
3
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1answer
40 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 ...
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1answer
69 views

When an Intersection of Prime Ideals is a Prime Ideal

Let $R$ be an arbitrary ring, $\{P_1,....,P_n\}$ be a set of prime ideals. Verify that $P_1 \cap ... \cap P_n$ is prime if and only if there exists $1 \leq i \leq n$ such that $P_i$ is contained in ...
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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 ...
3
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1answer
41 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 ...
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1answer
72 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 ...
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1answer
45 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 ...
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1answer
48 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 $\alpha:M'\rightarrow M$ ...
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2answers
147 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 ...
8
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1answer
71 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
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3answers
462 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 ...
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0answers
22 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 ...
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1answer
30 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
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2answers
74 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 ...
2
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0answers
20 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 ...
2
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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 ...
3
votes
1answer
109 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 ...