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
22 views

Fraction modulo integer in sage

I'm working on a sage script right now, I have some polynomials coefficients that are rational, and I want to apply a congruence on these coefficientss, for example: $p = 1 + (7/2)x$ the function ...
0
votes
0answers
6 views

UTMn[D] is artinian

Why is the upper triangular matrices over a division ring D is artinian? I tried to find properties of this class of rings. The only thing I found that the jacobson radical of this ring is the ...
4
votes
2answers
41 views

Matrix ring $M_2(\mathbb{C})$, $\mathbb{C}^2$ with $M_2(\mathbb{C})$-module structure.

Let $R$ be the $2 \times 2$ matrix ring $M_2(\mathbb{C})$. let $M = \mathbb{C}^2$ with its natural $R$-module structure (just given by the usual action of $2 \times 2$ matrices on $2$-dimensional ...
1
vote
1answer
12 views

Does “pseudo-independent implies independent” imply that $R$ is a field?

(All my rings are unital.) Suppose $R$ is a commutative ring and that $M$ is an $R$-module. Definition. Call a subset $X \subseteq M$ pseudo-independent iff for all proper subsets $Y$ of $X,$ the ...
2
votes
2answers
18 views

Cyclic projective module

Let $R$ be an integral domain. If $M$ is a cyclic $R$-module which is also projective, then must there necessarily be an isomorphism of $R$-modules $M \cong R$?
0
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0answers
23 views

Is $\oplus_{i \in I} M_i$ necessarily projective?

If $\{M_i\}$ is a collection of projective modules over a ring $R$, then must the direct sum $\oplus_{i \in I} M_i$ necessarily be projective? I understand that each direct sum of two modules is ...
2
votes
1answer
24 views

If $A \times B$ is the direct product of two rings, must $A$ necessarily be a projective $A \times B$-module?

If $A \times B$ is the direct product of two rings, must $A$ necessarily be a projective $A \times B$-module? I understand I am supposed to think of $A$ as an $A \times B$-module by identifying ...
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0answers
22 views

Show that $I + J = \{a + b\vert a \in I, b \in J\}$ is an ideal of R.

Let $I, J$ be ideals of a ring $R$. Show that $I + J = \{a + b\vert a \in I, b \in J\}$ is an ideal of R. Because $I,J$ are ideals of $R$, so $I,J$ both have $0$, thus $0+0=0\in I+J$. This shows ...
0
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0answers
20 views

How does one prove that $\mathbb{Z}[\sqrt{-41}]$ is a unique factorization domain.

By extending the Euclidean algorithm one can show that $\mathbb{Z}[i]$ has unique factorization. This logic extends to show $\mathbb{Z}[\frac{1 + \sqrt{-3}}{2}]$, $\mathbb{Z}[\frac{1 + ...
4
votes
2answers
106 views

Basic question about finite fields and characteristic

I am reading Herstein's "Topics in Algebra" and I've encountered with the following problem: If $D$ is an integral domain and $D$ is of finite characteristic, prove that the characteristic of $D$ is ...
0
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1answer
19 views

If $\phi(F)\neq \{0\}$, then $F\cong R$.

Let $F$ be a field and $R$ be a ring. Suppose $\phi:F\rightarrow R$ is a ring homomorphism. Show that if $\phi(F)\neq \{0\}$, then $F\cong R$. Suppose $R$ is a ring and $\phi: F\rightarrow R$ is a ...
9
votes
7answers
92 views

Why is $\mathbb{Z}[X]$ not a euclidean domain? What goes wrong with the degree function?

I know that $K[X]$ is a Euclidean domain but $\mathbb{Z}[X]$ is not. I understand this, if I consider the ideal $\langle X,2 \rangle$ which is a principal ideal of $K[X]$ but not of $\mathbb{Z}[X]$. ...
5
votes
3answers
65 views

examples of non-abstract rings?

In group theory, it helped me a lot to use symmetry groups of geometrical objects like a triangle to understand the more abstract concepts. Are there rings with a similar low level of abstractness? I ...
2
votes
2answers
36 views

Prove that $(5x^3+9x^2-27x+3)$ is a maximal ideal in $Q[x]$

We have shown that $Q[x]$ is a Euclidean Domain, and thus is a Principal ideal domain. A principal ideal $(f)$ is maximal $\iff$ $f$ is irreducible in $Q[x]$. But how do I show that \begin{equation*} ...
1
vote
2answers
51 views

Proof for quotient polynomial rings equivalent to field extension

I am predominantely looking for a proof, I have seen in my books and around but seem to have a hard time finding that if we let $\alpha_1,\alpha_2,...,\alpha_n$ be the roots of the minimal polynomial ...
0
votes
2answers
36 views

Field Extension Question for Polynomials

I cannot seem to find the answer to this question on the internet. It is a question about field extensions for an element $a,b \neq F$ but in some extension $K$. I am wondering if $F(a,b)= \lbrace ...
5
votes
1answer
45 views

Generated subring and finiteness

I need some help with this question: Let $A$ be the subring of $\mathbb{Q}(i)$ generated by $\mathbb{Z}[i]$, $\frac{1}{1+2i}$ and $\frac{1}{2+3i}$. Given $n\in\mathbb{Z} \setminus \{0\}$, can we ...
2
votes
1answer
59 views

Are fractions with zero divisors in the denominator never well defined?

Are fractions with zero divisors in the denominator never well defined? I know that for a fraction in modular arithmetic to be well defined, the denominator must not be a zero divisor, e.g: $$ x ...
2
votes
1answer
22 views

References for loop rings

I recently saw a paper on alternative loop rings, as always, I am interested in all kinds of rings, and new kind of ring looked attractive. I would like to read loop and then loop rings in detail from ...
0
votes
2answers
58 views

Applications of $\mathbb{Z}/n\mathbb{Z}$ [on hold]

I would like someone to proof me this claim and give me its applications in mathematics if it's not a convention. Claim: for all positive integers $n$, the ring $\mathbb{Z}/n\mathbb{Z}$ is domain if ...
1
vote
1answer
57 views

Exact functors preserve free modules?

Let $R$ be a principal ideal domain. Suppose that $F$ be an exact functor from the category of $R$-modules to itself. If $M$ is a free $R$-module, is $F(M)$ still a free $R$-module? I do not know how ...
5
votes
3answers
68 views

Multiplicative Inverse Element in $\mathbb{Q}[\sqrt[3]{2}]$

So elements of this ring look like $$a+b\sqrt[3]{2}+c\sqrt[3]{4}$$ If I want to find the multiplicative inverse element for the above general element, then I'm trying to find $x,y,z\in\mathbb{Q}$ such ...
2
votes
0answers
22 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 ...
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votes
1answer
24 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
87 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 ...
4
votes
2answers
29 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
18 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
54 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 $a = b = (2) \in \mathbb Z_6$
2
votes
2answers
58 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, ...
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 $R$. ...
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votes
2answers
37 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 ...
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
35 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 ...
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 ...
3
votes
2answers
76 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: $$a_nx^n+a_{n-1}x^{n-1}+\cdots+a_2x^2+a_1x+a_0 {\pmod {x-1}}$$ ...
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 ...
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 ...
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
40 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 ...
2
votes
1answer
37 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
53 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) ...
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 ...
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 ...
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 ...
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 ...