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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5
votes
3answers
636 views

Homomorphisms from a unital ring to a ring with no zero divisors preserve unity?

I'm having a bit of trouble with a problem from Hungerford's Algebra concerning ring homomorphisms. Let $f\colon R\to S$ be a homomorphism of rings such that $f(r)\neq 0$ for some nonzero $r\in ...
4
votes
1answer
156 views

General form of elements in $ \mathbb{Z} [\frac{1+\sqrt{-3}}{2} ] $

What is the general form of elements in $\displaystyle \mathbb{Z} \left[\frac{1+\sqrt{-3}}{2} \right] $? I'm getting muddled. Thanks
7
votes
3answers
2k views

The subring test

This is how the wikipedia article on subring defines the subring test The subring test states that for any ring $R$, a nonempty subset of $R$ is a subring if it is closed under addition and ...
7
votes
1answer
567 views

Ring homomorphisms which map a unit to a unit map unity to unity?

this is the third part of a question I've been working on from Hungerford's Algebra. It is exercise 15 in the first section of Chapter III. $(c)$ If $f\colon R\to S$ is a homomorphism of rings ...
4
votes
5answers
661 views

Why is the endomorphism ring of $\mathbb{Z}\times\mathbb{Z}$ noncommutative?

So I hear that the endomorphism ring of an abelian group is not always commutative. In particular, I'm looking at the abelian group $A=\mathbb{Z}\times\mathbb{Z}$, and considering $\text{End } A$. I ...
0
votes
1answer
59 views

Prove a property of when simple modules are isomorphic

Let $\mathfrak{m}_1$ and $\mathfrak{m}_2$ be left maximal ideals of a unital ring $A$. Show that the simple modules $A/\mathfrak{m}_1$ and$A/\mathfrak{m}_2$ are isomorphic if and only if there exist ...
5
votes
4answers
838 views

Lack of unique factorization of ideals

I'm aware of the result that integral domains admit unique factorization of ideals iff they are Dedekind domains. It's clear that $\mathbb{Z}[\sqrt{-3}]$ is not a Dedekind domain, as it is not ...
4
votes
4answers
780 views

Proving that an ideal in a PID is maximal if and only if it is generated by an irreducible

Let $R$ be a PID (Principal Ideal Domain) and $x$ is an element R. Prove that the ideal $\langle x\rangle$ is maximal if and only if $x$ is irreducible. Ok, so I know what an irreducible is. I'm ...
4
votes
1answer
242 views

Lifting maps of quotient modules

Today I tried to check this, but couldn't see how to do it. I think it is probably a standard result, but a brief check of Atiyah-Macdonald didn't yield anything, and I don't know what to google for. ...
5
votes
1answer
147 views

Lang's “General Integrality Criterion”

Theorem 3.7 in the chapter on ring extension on page 352 of the latest edition of Lang's "Algebra" appears redundant in its phrasing to me. Specifically, if $g_s$ is a polynomial of total degree ...
1
vote
2answers
220 views

Why is $R[X,Y]/(X^2-Y^3)$ isomorphic to $\{\sum a_iT^i\in R[T] \; : \; a_1=0\}$?

Let $R$ be a ring (commutative, with unit). Show that $A=\{\sum a_iT^i\in R[T] \; : \; a_1=0\}$ is a subring of $R[T]$ and isomorphic to $R[X][Y]/(X^2-Y^3)$. Of course, I'm trying to find a ring ...
4
votes
3answers
550 views

Dedekind's theorem on the factorisation of rational primes

Let $K$ be an algebraic number field, and suppose its ring of integers is $\mathcal{O}_K = \mathbb{Z}[\theta]$ for some $\theta \in \mathcal{O}_K$. Let $f \in \mathbb{Z}[X]$ be the minimal polynomial ...
0
votes
2answers
262 views

Is there a better way of writing this ring?

The quotient $k[x,y]/(x-y^2)$ is isomorphic to $k[y]$ as a ring. Suppose, $g$ is a polynomial in $y^2$. Is there a "nice" ring that is isomorphic to $k[x,y^2]/(x^2-gy^2)$ assuming $g$ is not a unit? ...
5
votes
3answers
292 views

When is $\mathrm{Hom}(A,R) \otimes B =\mathrm{Hom}(A,B)$?

It is simple to show that if $A,B$ are vector spaces, then $A^* \otimes B = \text{Hom}(A,B)$, where $A^*$ is the dual of $A$. To what extent does this hold for modules, where we should interpret $B^*$ ...
0
votes
1answer
60 views

$\mathbb{C}$-vector space structure inside nontrivial f.g. module over $M_2 (\mathbb{C})$ has dim $\geq 2$

I'm trying to prove: If $M$ is a nontrivial finitely generated left module over $M_2 (\mathbb{C})$, then the accompanying $\mathbb{C}$-vector space structure (just restrict the action to the scaling ...
8
votes
5answers
1k views

Non-unital rings: a few examples

Every ring I've ever heard of is unital, i. e., contains a (unique) element $a$ such that $xa = ax = x$ for every $x$ in it. However, some rings do not have such an element. What are they? P. S.: one ...
2
votes
0answers
99 views

Geometric understanding of principal/non-principal ideals

A number field $K$ with the $r$ embeddings into $\mathbb R$ and $2s$ pairs of conjugate embeddings into $\mathbb C$ can put into ring homomorphism with the product of rings $\mathbb R^r \times \mathbb ...
2
votes
1answer
333 views

Methods of determining if a non-primitive polynomial is irreducible in a ring

Take, for example, $ X^4 + 2X + 2 $ in $ \mathbb{Q}[X] $. How do I determine if this is irreducible? Thoughts: I know Gauss' Lemma and Eisenstein's criterion, but they only work for primitive ...
3
votes
2answers
889 views

Show that if $ R $ is an integral domain then a polynomial in $ R[X] $ of degree $ d $ can have at most $ d $ roots

Show that if $ R $ is an integral domain then a polynomial in $ R[X] $ of degree $ d $ can have at most $ d $ roots. Thoughts so far: I feel like I might be missing something here. If $ R $ is an ...
3
votes
1answer
221 views

Let $ n \geq 3 $. By factorising $ n $ or $n + 1 $ (as appropriate), show that $ \mathbb{Z}[\sqrt{-n}] $ is not a UFD

Let $ n \geq 3 $. By factorising $ n $ or $n + 1 $ (as appropriate), shat that $ \mathbb{Z}[\sqrt{-n}] $ is not a UFD. My thoughts so far: Define $ N(a + b \sqrt{-n}) = a^2 + n b^2 $. Suppose $ n $ ...
5
votes
5answers
559 views

Exhibit an integral domain $R$ and a non-zero non-unit element of $R$ that is not a product of irreducibles.

Exhibit an integral domain $R$ and a non-zero non-unit element of $R$ that is not a product of irreducibles. My thoughts so far: I don't really have a clue. Could anyone direct me on how to think ...
4
votes
1answer
407 views

Give an element of $ \mathbb{Z}[\sqrt{-17}] $ that is a product of two irreducibles and also a product of three irreducibles

Give an element of $ \mathbb{Z}[\sqrt{-17}] $ that is a product of two irreducibles and also a product of three irreducibles. My thoughts so far: Using the multiplicative norm $ N(a + b\sqrt{-17}) = ...
5
votes
4answers
1k views

Show that the ideal $ (2, 1 + \sqrt{-7} ) $ in $ \mathbb{Z} [\sqrt{-7} ] $ is not principal

Show that the ideal $ I = (2, 1 + \sqrt{-7} ) $ in $ \mathbb{Z} [\sqrt{-7} ] $ is not principal. My thoughts so far: Work by contradiction. Assume that $ I $ is principal, i.e. that it is generated ...
3
votes
2answers
100 views

equality of modules

I'm reading a proof of Nakayama's theorem; it says at a certain step that: For $M$, a finitely generated module on a ring $R, N$ a submodule, and $I$ an ideal of the ring $R$: If $M = N + IM$, then ...
4
votes
2answers
259 views

Left Artinian Rings

I have an attempt to prove the claim that $R$ has finitely many nonisomorphic simple $R$-modules if $R$ is left artinian. I would like to know if it's a good attempt. Helpful hints are very much ...
1
vote
4answers
1k views

Abstract Algebra ring homomorphism

Let $\phi\colon R \to R'$ be a ring homomorphism. Prove that if $R$ is a field then either $R$ is an isomorphism or $\phi(r) = 0$ for all $r \in R$. I am stuck on this problem and don't know ...
1
vote
1answer
65 views

Submodules of $A\times B$

I am self studying ring theory and modules from Rotman's Advanced Modern Algebra. I would like some help on putting this thought to bed. Let $A$ and $B$ be rings. Let $R=A\times B$. Is it ...
3
votes
1answer
162 views

Semisimple rings

Let R and S be rings. Show that R x S is semisimple if and only if both R and S are semisimple. I have the converse direction ($\Leftarrow$). It is the other direction ($\Rightarrow$), that I ...
14
votes
6answers
3k views

Show that $\langle 2,x \rangle$ is not a principal ideal in $\mathbb Z [x]$

Hi I don't know how to show that $\langle 2,x \rangle$ is not principal and the definition of a principal ideal is unclear to me. I need help on this, please. The ring that I am talking about is ...
4
votes
1answer
281 views

Some isomorphism conditions

Here I want to ask some true or false problems regarding isomorphisms (and if false, is there some extra condition to make it true). I do not what is the correct title for these problem. And I also ...
2
votes
0answers
132 views

Exterior algebras and radicals

So without giving excessive background, I was working on baby Spivak & got to the stuff about exterior algebras, & now I'm going through the section on tensor algebras in my copy of ...
15
votes
1answer
299 views

Does a finite ring's additive structure and the structure of its group of units determine its ring structure?

Let $A$ and $B$ be finite commutative rings with unity. Denote the additive group structure of each to be $A^{(+)}$ and $B^{(+)}$, and the multiplicative group of units of each to be $A^{(\times)}$ ...
10
votes
2answers
274 views

Minimal systems of generators for finitely generated algebras over commutative (graded) rings

Let $S$ be some base ring (a commutative ring or even just a field), and $R$ a commutative ring containing $S$ which is finitely generated (as an algebra) over $S$. What conditions guarantee that ...
1
vote
0answers
60 views

Computing relations on the columns of a matrix

Given an $m\times n$ (with $n>m)$ matrix $M$ over a polynomial ring $R=k[x_1,...,x_n]$, suppose that every column of $M$ is an $R$-linear combination of $m$ specified columns. I would like to ...
4
votes
2answers
304 views

Two different ideals with the same annihilator

Is this statment always true? $$(a)\subsetneq (b)\Rightarrow \text{Ann}_R b\subsetneq \text{Ann}_R a$$ If it is false, can you please provide an example? Also what is the largest class of rings that ...
2
votes
3answers
156 views

Simplifying expressions

I have a polynomial ring $R=k[x,y,z...]$ and a given ideal $I$ (defined by given generators) and several polynomials $f_1,f_2,...$ in the ring. I also have several other elements of $R$ given as ...
6
votes
3answers
396 views

Does this give a localization of a ring?

Apologies in advance for the naivety of this question. Let $R$ be a commutative (resp. non-commutative) ring, $S \subset R$, and let $R' = R[x_s (s \in S)]$ be the polynomial ring obtained by ...
7
votes
2answers
501 views

Why are Dedekind-finite rings called so?

A Dedekind-finite ring is a ring in which $ab=1$ implies $ba=1$. It seems natural to look for a connection to Dedekind-finite sets, however for such a set any injective endomorphism is surjective, ...
2
votes
1answer
278 views

Name of a book with the following contents?

Some time ago, I received Algebra notes from my advisor who is advising me on a project. I learned very much from these notes and wondered what the name of the book from which these notes were ...
3
votes
1answer
200 views

How to prove that every algebraically closed integral domain is a field?

Suppose $R$ is an integral domain and $R$ is algebraically closed. Prove that it then follows $R$ is a field.
3
votes
2answers
124 views

What is useful about commutativity/non-commutativity in practice

How is having commutativity or not having it useful for in practice? (whether it is for linear algebra, rings, basic airthmetic etc)
5
votes
2answers
244 views

For which prime $p$ are the additive groups $\mathbb{F}_p$ and $\mathbb{F}_p^2$ $\mathbb{Z}[i]$-modules?

Homework for my algebra class. Chapter 14, Exercise 7.8 in Artin's Algebra, Second Edition: Let $F = \mathbb{F}_p$. For which prime integers $p$ does the additive group $F^1$ have a structure ...
6
votes
3answers
802 views

Localization of a prime ideal in $\mathbb{Z}/6\mathbb{Z}$

How can we compute the localization of the ring $\mathbb{Z}/6\mathbb{Z}$ at the prime ideal $2\mathbb{Z}/\mathbb{6Z}$? (or how do we see that this localization is an integral domain)?
11
votes
2answers
1k views

Inverse Image of Maximal Ideals

Given a map of commutative rings with unit, it is often the case that the inverse image of a maximal ideal is not maximal. For example, consider the inclusion $\mathbb{Z} \subseteq \mathbb{Q}$. ...
0
votes
1answer
100 views

What is the type of the “adjoin” operator on rings?

Context: I am trying to formalize the definition of a polynomial ring in a programming language. One way to think about it, is to start by formulating the definition of a ring extension by adjoining ...
1
vote
1answer
209 views

How to find matrices with given commutator

Consider $M_2(\mathbb{Z})$. Is it possible to find two matrices A,B such that their commutator AB - BA equals a given matrix C? Is there any chance to characterize all possible occuring commutators in ...
1
vote
1answer
161 views

Why is IBN not a Morita property?

If a ring $R$ satisfies the property $P$, but the matrix rings $M_n(R)$ do not satisfy $P$, then $P$ is not a Morita property; that is the definition. But that is not fix this situation. I want to ...
1
vote
1answer
230 views

Automorphisms of $\mathbb{R}^n$

It is well known that the only ring automorphism of $\mathbb{R}$ is the identity. This follows from the fact that all ring automorphisms of $\mathbb{R}$ must fix $0$, and be order preserving, and ...
25
votes
3answers
944 views

A finite ring is a field if its units $\cup\ \{0\}$ comprise a field of characteristic $\ne 2$

Suppose $R$ is a finite ring (commutative ring with $1$) of characteristic $3$ and suppose that for every unit $u \in R\:,\ 1+u\ $ is also a unit or $0$. We need to show that $R$ is a field. Is this ...
1
vote
1answer
84 views

Polynomials f(x) of degree at most 5 forming a ring and field

Show that the set of all polynomials f(x) of degree at most 5 with integer coefficients is a ring. Is the set of such polynomials a field? I don't see how the ring of polynomials with degree at most ...