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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2
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0answers
14 views

Reference Request: Replacement for Anderson and Fuller

I'm working through Rings and Categories of Modules by Anderson and Fuller. It's a great text, but I would like a more modern replacement. The text focuses too much on the language of generation and ...
-4
votes
1answer
23 views

Find an example of a ring with nonzero unity,1, that has a subring with nonzero unit 1" that is not equal to 1. [on hold]

This problem is from Fraleigh "Abstract Algebra". The hint said to consider a direct product or Z_6.
2
votes
3answers
84 views

Example of commutative ring that doesn't satisfy distribution of intersection over addition

I'm trying to find an example of commutative ring $R$ and ideals $\mathfrak a,\mathfrak b,\mathfrak c \in R$ such that $$\mathfrak a \cap (\mathfrak b + \mathfrak c) \neq \mathfrak a \cap ...
3
votes
1answer
69 views

Nakayama's lemma, second version

Let $R$ be a commutative ring with identity, $J$ an ideal that is contained in every maximal ideal of $R$, and $A$ is finitely generated $R-$ module. If $R/J\otimes _R A=0$, then $A=0$. ...
3
votes
1answer
22 views

Ring with no identity (that has a subring with identity) has zero divisors.

Let $L$ be a non-trivial subring with identity of a ring $R$. Prove that if $R$ has no identity, then $R$ has zero divisors. So I assumed that there $\exists$ $e \in L$, such that $ex=xe=x$, ...
-1
votes
1answer
15 views

Please explain how should I go about proving a domain is not integrally closed. [on hold]

In particular, I need to prove $\mathbb{Z} [i\sqrt{3}]$ is not integrally closed.
0
votes
1answer
52 views

Wrong proposition in “Atiyah and Macdonald”s book?!

In page 6 of "Introduction to commutative algebra" says that: $a \cap b = ab$ provided $a + b = (1)$ But i think it's not true,by considering $a = b = (2) \in \mathbb Z_6$
0
votes
1answer
27 views

Cardinality of Quotient ring

Let $R$ be the ring obtained by taking the quotient of $\mathbb{Z_6}[X]$ by principal ideal $(2x+4)$. Then 1) $R$ has infinite elements 2) $R$ is field 3) $5$ is unit in $R$ 4) $4$ is unit in $R$. ...
2
votes
2answers
57 views

How to workout what elements of a quotient ring look like?

I am trying to understand quotient rings. Firstly: $$\frac{\Bbb Z[x]}{\langle x-1\rangle}$$ The above I can understand in a fairly naive way. Since the ideal is generated by a degree one polynomial, ...
-5
votes
2answers
34 views

Show $R/I$ is a ring with unity,$1 + I$ [on hold]

Suppose $R$ is a ring with unity and $I \neq R $ is an ideal of $R$. Show that $R/I$ is a ring with unity,$1 + I$ . Can anyone give me a hit to do this question? Thanks
0
votes
1answer
33 views

Prime ideal in Dedekind ring is finitely generated

Let $R$ be a Dedekind ring, which means integral domain, integrally closed, Noetherian, which means that given any chain of ideals in $R$: $$I_1\subseteq\cdots \subseteq I_{k-1}\subseteq ...
1
vote
1answer
39 views

About right identity which is not left identity in a ring

Let $S$ be the subset of $M_2(\mathbb{R})$ consisting of all matrices of the form $\begin{pmatrix} a & a \\ b & b \end{pmatrix}$ The matrix $\begin{pmatrix} x & x \\ y & y ...
4
votes
0answers
44 views

Prime ideals in a quotient

I am interested in finding the number of prime ideals in $\mathbb{Z}[x]/(12,x^2+1)$. Here is what I think. Modding out by $x^2+1$, we get $\mathbb{Z}[i]/(12)$. Factoring $12$ in the Gaussian ...
0
votes
1answer
21 views

How does this prove that the ring homomorphism is surjective?

The course notes on rings have the below lemma Let $R$ be a ring and $I$ a two sided ideal. Define $\pi : R \rightarrow R/I$ by $\pi(r)=r+I$. Then $\pi$ is a surjective ring map and ker $\pi=I$. ...
0
votes
1answer
34 views

$I$ is a two sided ideal in $A$, $L$ and $M$ are $A$-modules such that $L \subset M$. If $l\in L$ and $l\in MI$ then $l \in LI$?

$A$ is an Artinian ring, $I$ is a two sided ideal. I know that $L \subset M$ are $A$-modules, and that $l \in L \cap MI$. Is it true that $l \in LI$? If is not true, can I have an example where it ...
0
votes
2answers
138 views

Proving that the intersection of two subrings of R is also a subring of R

If $R_1$ and $R_2$ are both subrings of $R$ , how to prove that $R_1 \cap R_2$ is also a subring of $R$. here is my attempt (1) since $R_1$ is a subring of $R$ then it must contain zero (identity ...
10
votes
1answer
124 views

DVR, power series expansion.

Let $A$ be a discrete valuation ring with quotient field $K$, maximal ideal $\mathfrak{m}$, uniformizing parameter $t$. Let $k = A/\mathfrak{m}$, so $k$ is a field. How do I show that there is a ...
1
vote
0answers
19 views

Why are principal fractional ideal also fractional ideals?

I don't understand the following: Let $K$ be the quotient field of an integral domain $R$. A fractional ideal $I$ is a subset of $K$ other than $\{0\}$, for which a $0\neq r\in R$ exists, so that ...
-1
votes
3answers
58 views

Example of ideal generated by two elements

I have an easy example on my notes that I don't understand. My teacher said that in $\mathbb{Z}$, $(2,3)=2\mathbb{Z}+3\mathbb{Z}$ is a principal ideal, because $2\mathbb{Z}+3\mathbb{Z}=\mathbb{Z}$. ...
3
votes
2answers
75 views

Show that the rings $2\mathbb{Z}$ and $3\mathbb{Z}$ are not isomorphic.

Here I am under the impression that $2\mathbb Z$ and $3\mathbb Z$ are the sets of even numbers and multiples of $3$ respectively and the operations are usual addition and multiplication. This is an ...
0
votes
1answer
58 views

Equivalence between category of $R$-modules and $S$-modules

Is it true that by equivalence between category of $R$-modules and $S$-modules we conclude that $R$ and $S$ are isomorphic?($R$ and $S$ are unital rings.)
2
votes
1answer
29 views

Arbitrary elements in a quotient ring $\Bbb R[x]/(x-1)$

If I have an ideal $(x-1)$ for the ring $\Bbb R[x]$, how do I think of the quotient ring $\Bbb R[x]/(x-1)$? I have all polynomials with: $$a_nx^n+a_{n-1}x^{n-1}+\cdots+a_2x^2+a_1x+a_0 {\pmod {x-1}}$$ ...
3
votes
2answers
47 views

Understanding Quotient Rings

I am watching a video on Field Extensions (trying to self "relearn" some Abstract Algebra before I take it again). I struggled with it as an undergrad, so I'm trying to get a leg up. The example ...
2
votes
1answer
37 views

Zeroes of prime polynomials in the algebraic torus (A Hilbert's Nullstellensatz for Laurent polynomials?)

Let $Q\in\mathbb C[z_1,\dots,z_D]$ be a prime polynomial and let $Z(Q)$ be the algebraic hypersurface of its zeroes. Assume that $P\in\mathbb C[z_1,\dots,z_D]$ is a polynomial which has zeroes at ...
7
votes
0answers
62 views
+100

Example of a commutative, local, dual ring with nilradical $N$ such that $ann(N)\subsetneq N$

For an ideal $I\lhd R$ in a commutative ring $R$, let $ann(I)$ denote the annihilator of $\{x\in R\mid xI=\{0\}\}$. A commutative ring $R$ is said to be a dual ring if for every ideal $I$ of $R$, ...
0
votes
2answers
38 views

How to go about this proof for non zero polynomials.

How do I go about proving this? Let $\mathbb{F}$ be a field and $X$ an indeterminate, and consider the polynomial ring $\mathbb{F}[X]$. Let $f(X), g(X) \in \mathbb{F}[X]$ with $f(X), g(X) \neq ...
2
votes
1answer
36 views

Finite covering by finitely generated B-algebras

While I was working on the first properties of schemes on Hartshorne's Books, I needed the following result that I couldn't prove it: Let $X=Spec(A)$ be an affine scheme. Suppose that there exist an ...
3
votes
1answer
39 views

Algebraic closure for rings

Is there any notion of algebraic closure for commutative rings? I am specifically interested in such a concept for $\mathbb Z_n$, with $n$ not a prime (possibly square-free). Such a concept would be ...
1
vote
2answers
51 views

prove $(m) \subset (n)$ iif $n$ divides $m$

For non-zero integers $m$ and $n$, prove $(m) \subset (n)$ iif $n$ divides $m$, where $(n)$ is the principal ideal. My attempt is following. For non-zero integers $m$ and $n$, assume that $(m) ...
11
votes
3answers
728 views

Are there more groups than rings?

It seems pretty clear to me that both of these are at least uncountable (which I think I could prove with some work). It also seems that you should be able to make some diagonal argument about the ...
2
votes
2answers
38 views

A question on Join homomorphism and Ideals

On page 287 of the book Mathematical Methods in Linguistcs, by Barbara Partee, Alice Ter Meulen and Robert E. Wall (Dordrecht, Kluwer Academic Press, 1993), I find the following theorem, which they ...
2
votes
2answers
59 views

Proving $0x=0$ in a ring

I am trying to prove the above trivial statement. I am aware of the standard approach of letting $0 = 0 + 0$ and cancelling, but I would like the below statement to be verified/corrected: $1\cdot ...
2
votes
1answer
34 views

About the ways prove that a ring is a UFD.

I'm doing an exercise where I have a commutative ring with unity $R$. We had to find that the nonunits formed an ideal (maximal). After that, we found the irreducible elements, and then we saw that ...
1
vote
1answer
26 views

Polynomial Rings

I know that an Euclidean ring is a principal ring and this last is a factorial ring. I also know that if $B$ is a field, then $B[x]$ is Euclidean. While, for instance $\mathbb{Z}[x]$ is not a ...
0
votes
1answer
46 views

What does the ring $R=C[x]/I$ look like?

Maybe it's a stupid question but what does the ring $R=C[x]/I$ look like? $I$ is the ideal in $C[x]$. Everything helping! Thanks :)
3
votes
1answer
366 views

Correspondence theorem for rings.

Could someone provide a reference that includes a full and honest proof of the Correspondence Theorem for rings? Let $A$ be a multiplicative ring with identity and $I$ an ideal of $A$. There is a ...
2
votes
1answer
33 views

If some non-primitive polynomial in $\mathbb Z[x]$ is irreducible over $\mathbb Q$, does this imply it is irreducible over $\mathbb Z$?

I know from a lemma in Herstein that if a primitive polynomial in $R[x]$ is irreducible in its field of quotients $F[x]$, then it is irreducible in $R[x]$. But, if some non-primitive polynomial in ...
5
votes
1answer
47 views

Only DVR's with quotient field $\mathbb{Q}$?

Let $p \in \mathbb{Z}$ be a prime number. I know how to show that $$\{r \in \mathbb{Q}: r = {a\over{b}},\text{ }a,b \in \mathbb{Z},\text{ }p\text{ doesn't divide }b\}$$ is a DVR with quotient field ...
2
votes
0answers
84 views

Regular subrings of a polynomial ring

Let $R=\mathbb{C}[x,y]$. I have the following situation: $\mathbb{C} \subseteq D \subseteq R$ is affine (=finitely generated as a $\mathbb{C}$-algebra), noetherian, has field of fractions ...
3
votes
2answers
48 views

about center of group rings $RG$ and $(R/I)G$

Let $I$ be an ideal of a ring $R$. It is mentioned in the book An Introduction to Group Rings (by Sehgal and Milies) that the canonical homomorphism $RG \rightarrow (R/I)G$ maps $Z(RG)$, center of ...
5
votes
1answer
57 views

$R$ is Finite ring and for every $a \in R$, there exist natural number $n(a)$ ST $a^{n(a)}=a$

$R$ is Finite ring and for every $a\in\,R$ there exist natural number $n(a)>1$ that $a^{n(a)}=a$ . Is $R$ is a ring with identity? If this question is correct then, for every $a \in ...
4
votes
1answer
27 views

Local ring coincides with DVR.

Assume $A$ is a discrete valuation ring with quotient field $K$ and maximal ideal $\mathfrak{m}$. If $S$ is a local ring containing $A$ and contained in $K$ with maximal ideal containing ...
2
votes
1answer
92 views

$R$ is a commutative integral ring, $R[X]$ is a principal ideal domain imply $R$ is a field

I've just read a proof of the statement: Let $R$ be a commutative integral ring. If $R[X]$ is a principal ideal domain, then $R$ is a field. In one part of the proof there is a step which I ...
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votes
0answers
84 views

Properties of a localization of $\mathbb Z$

Let $R=\{\frac ab ∈ ℚ ∣ b \text{ is odd}\}$. (1) Prove that $R$ is isomorphic to $ℤ_P$, where $ℤ_P$ is the localization at $P$, for a prime ideal $P$ of $R$. (ii) Find $U(R)$. Prove that ...
1
vote
0answers
18 views

If $I$ is a finitely generated semisimple left $R-$module, show that $I=Re$ for an idempotent $e \in I$.

Let $I$ be a left ideal in a semiprime ring $R$. If $I$ is a finitely generated semisimple left $R-$module, show that $I=Re$ for an idempotent $e \in I$.
4
votes
4answers
114 views

Show that the ideal of $k[X_1, X_2, X_3]$ generated by $X_1^3-X_3$ and $X_2^2-X_3$ is a prime ideal.

Show that the ideal $I$ of $k[X_1, X_2, X_3]$ generated by $X_1^3-X_3$ and $X_2^2-X_3$ is a prime ideal, where $k$ is a field. I tried to prove it by contradiction. Suppose $f$ and $g$ are not of ...
1
vote
2answers
120 views

Let $R=\{a/b \in \mathbb{Q} \mid \text{$b$ is odd}\}$

Let $R=\{a/b \in \mathbb{Q} \mid \text{$b$ is odd}\}$. Prove that $R$ is isomorphic to $\mathbb{Z}_P$, where $\mathbb{Z}_P$ is the localization at $P$, for a prime ideal $P$ of $R$. I really ...
1
vote
1answer
32 views

Regularity of a quotient ring of the polynomial ring in three indeterminates

Let $I=(f)$ be a prime ideal in $R=\mathbb{C}[x,y,z]$, so $f$ is an irreducible polynomial, and further assume that $f$ is of the following form: $f=z^n+c_{n-1}z^{n-1}+\ldots+c_1z+c_0$, where ...
3
votes
1answer
82 views

Question about ideals of a ring: $I\cdot J=I \implies J=I$?

Doing exercises, this question came to my mind. Is it true that if $I$ and $J$ are proper and nonzero ideals of a ring $R$, $$I\cdot J=I \implies J=I?$$ And $$I\cdot J=I \iff J\subseteq I?$$
6
votes
2answers
98 views

Maximal left ideals $\leftrightarrow$ simple left modules

Suppose $R$ is a ring with unity. This passage in Lang's Algebra discusses the correspondence $$\text{Maximal left ideals of $R$} \leftrightarrow \text{Simple left $R$ modules},$$ where I corresponds ...