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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2answers
84 views

$ I(J+L)=IJ+IL$ if $I,J,L$ are ideals of $K$

Given that $I,J,L$ are ideals of $K$, do we have $I(J+L)=IJ+IL$? I am confused how to do it.
1
vote
3answers
338 views

Show that if $R$ is an integal domain, then $R[X]$ is an integral domain.

Let $R$ denote an integral domain, and $R[X]$ denote the polynomials over $R$. Show that $R[X]$ is an integral domain. All I've got left is the non-trivial part - i.e. the cancellation property of ...
1
vote
2answers
158 views

Is $Z(R)$ a maximal ideal?

If $p$ and $q$ are two maximal ideals in the set of zero-divisors in a ring $R$ with non-zero intersection between $p$ and $q$. does the set of all zero-divisors are a maximal ideal and equal the ...
1
vote
6answers
670 views

$ p^{\frac1n} $ is irrational if $p $ is prime and $n>1$ [closed]

How can we prove that $ p^{\frac1n} $ is irrational if $p $ is prime and $n>1$?
0
votes
1answer
43 views

Why $(M/M \operatorname{rad} A) \operatorname{rad}A=0$?

Let $A$ be a ring and $M$ a right $A$-module. Why we have $(M/M \operatorname{rad}A) \operatorname{rad}A=0$? Thank you very much.
2
votes
1answer
30 views

Suppose $R$ is a ring containing a field $F$ in its centre. Construct an injective ring homomorphism from $R$ to $M_n(F)$

Let $R$ be a ring containing $F$ in its center. $R$ is an n-dimensional vector space over $F$, and the homomorphism is to be constructed in terms of a basis of $R$. I'm at a complete loss at what to ...
3
votes
1answer
52 views

How to show that $\operatorname{rad} (M \oplus N) = \operatorname{rad} M \oplus \operatorname{rad} N$?

Let $M, N$ be right $A$-modules. How can we show that $\operatorname{rad} (M \oplus N) = \operatorname{rad} M \oplus \operatorname{rad} N$?
3
votes
1answer
92 views

Question about radical of a module.

Let $M$ be a right $A$-module. How to show that $m\in \operatorname{rad}(M)$ iff for any simple right $A$-module $S$ and any $f\in \operatorname{Hom}_A(M, S)$, $f(m)=0$? I think that if $m$ is ...
8
votes
3answers
367 views

Using Hensel's Lemma to Factor a Polynomial over $\mathbb{Z}_4[x]$

We recently learned about codes over $\mathbb{Z}_4$, and Hensel's Lemma. The lemma is as follows: Let $f(x) \in \mathbb{Z}_4[x]$. Suppose $\mu(f(x)) = h_1(x)h_2(x) \cdots h_k(x)$, where $h_1(x), ...
2
votes
1answer
120 views

which of the following statements are true and why?

which of the following statements are true and why? Any two irreducibles in any UFD are associates. If $D$ is a PID, then $D[x]$ is a PID. In any UFD, if $p|a$ for an irreducible $p$, then $p$ ...
2
votes
1answer
119 views

Euclidean domain in which the quotient and remainder are always unique

Let $R$ be a Euclidean domain in which the quotient and remainder are always unique. Does it follow that the ring $R$ is either a field or a polynomial ring $F[X]$ for some field $F$?
2
votes
4answers
85 views

Is $\mathbb{Z}_{3}[X]/(X^2+X+1)$ a Euclidean domain?

Is $\mathbb{Z}_{3}[X]/(X^2+X+1)$ a Euclidean domain? If it is not, is it a principal ideal domain? Many thanks
2
votes
2answers
211 views

Algebraic Independence of Equations vs Polynomials

I am considering the difference between algebraic independence of a system of equations and polynomials. Are these two notions equivalent? For example, for $x, y, z$ real, $xy = A$ $yz = B$ $xz = ...
3
votes
3answers
169 views

How do I prove the lattice theorem for rings?

Let $ f: R \rightarrow S$ be an onto homomorphism from a ring $R$ to a ring $S$. Prove that there is a one-to-one, order-preserving correspondence between the ideals of $S$ and the ideals of $R$ that ...
4
votes
3answers
122 views

Identifying the quotient $\mathbb{R}[x]/(x^3+x)$ with a standard ring.

The question is in the title. By the first isomorphism theorem, I know that if I can find a surjective ring homomorphism $\varphi : \mathbb{R}[x] \rightarrow S$, where $S$ is some standard ring, and ...
7
votes
2answers
232 views

Why is the quotient map $SL_n(\mathbb{Z})$ to $SL_n(\mathbb{Z}/p\mathbb Z)$ is surjective?

Recall that $SL_n(\mathbb{Z})$ is the special linear group, $n\geq 2$, and let $q\geq 2$ be any integer. We have a natural quotient map $$\pi: SL_n(\mathbb{Z})\to SL_n(\mathbb{Z}/q).$$ I remember that ...
3
votes
1answer
231 views

Relation between localization and colimit.

I am trying to show that $S^{-1}R=\operatorname{colim}F(s)$, where $S$ is a multiplicative closed set in a commutative ring $R$ and $F$ is a functor from a filtered category $I$ to mod-$R$ and $I$ is ...
1
vote
1answer
155 views

Properties of ring of all holomorphic functions

Is $H(\mathbb C)=$, the ring of all holomorphic functions in $\mathbb C$, an UFD? and what are the irreducible and prime elements in it? answer: if $f(z)= z-a$ where $a \in \mathbb C$ then $f(z)$ ...
-2
votes
2answers
425 views

How many prime ideals does $\mathbb Q[x]/(x^m -1)$ have? (multiple choice)

Let $m$ be a positive integer, and $a_m$ denote number of distinct prime ideals of $\mathbb Q[x]/(x^m -1)$. Then which of the following are true? $a_4=2$ $a_4=3$ $a_5=2$ $a_5=3$
3
votes
1answer
790 views

For which $d$ is $\mathbb Z[\sqrt d]$ a principal ideal domain?

Is there any general idea about for which $d$, $\mathbb Z[\sqrt d]$ a principal ideal domain (PID)? As for example $\mathbb Z[\sqrt{-1}]$ and $\mathbb Z[\sqrt 2] $ are PIDs, but $\mathbb Z[\sqrt{-5}] ...
3
votes
1answer
139 views

Factorize $(9+11\sqrt{-5})$ as a product of prime ideals in $\mathcal{O}_K$ where $K=\mathbb{Q}(\sqrt{-5})$

The ring of integers in this case, is $\mathbb{Z}[\sqrt{-5}]$. I have calculated that the norm of $(9+11\sqrt{-5})$ is $686=2\times 7^3$ and therefore its prime factorization must contain a prime ...
8
votes
4answers
567 views

A question on definition of field of fractions

Wikipedia defines the field of fractions of a domain as The field of fractions or field of quotients of an integral domain is the "smallest" field in which it can be embedded. What does ...
4
votes
1answer
244 views

Defining multiplication on a Koszul complex

Let $R$ be a Noetherian commutative ring and $x$ and $y$ two elements in $R$. We construct the Koszul complex on $x$ and $y$. We start by the following two chain complexes: $$ C_2=0\to ...
4
votes
2answers
91 views

$\mathbb Z_m$ where every unit is an involution

What are all $m \in \mathbb N_{\geq 2}$ such that $\forall a \in (\mathbb Z_m^*): a^2 \equiv_m 1$? Hints would be nice :) This is not homework and question 2.24 in "Introduction to Algebra" from J. ...
4
votes
3answers
380 views

Examples of Morita equivalent rings

Can someone give some examples of Morita equivalent rings different from the classical one? (i.e. that a ring $R$ is Morita equivalent to the ring $M_n(R)$)
7
votes
1answer
855 views

For which $d<0$ is $\mathbb Z[\sqrt{d}]$ an Euclidean Domain? [duplicate]

I know that for $d=-1, -2$ the ring $\mathbb Z[\sqrt{d}]$ is an Euclidean Domain. I believe that it is not an Euclidean Domain for and $d \leq-3$. I have been able to prove it for a handful of ...
0
votes
2answers
161 views

Example finitely generated ideal by non-irreducible elements

Let $R$ be a ring (it doesn't need to be commutative nor unitary). I wonder if any finitely generated maximal ideal of $R$ must be generated by irreducible elements. I haven't been able to prove it ...
3
votes
1answer
126 views

Can An Axiom Schema be Independent?

Consider the following theory: Ring Theory (RT) + $\forall x(Sx=x+1)$ + first order induction (Ind). The finite rings $Z/nZ$ are models of this theory. Now consider RT + $\forall x(Sx=x+1)$ + ...
5
votes
6answers
2k views

Abstract algebra book recommendations for beginners.

I am taking abstract algebra in this semester. I find that my lecturer didn't provide enough examples for understanding. So I would like to ask what are the recommended books which has lots of solved ...
5
votes
1answer
135 views

Noncommutative Hilbert basis theorem is false?

How can I show that for a field $K$, in the free algebra on $2$ generators $K\langle x,y\rangle$, the two-sided ideal $$\big\langle\!\big\langle xy^ix\;\big|\;i\in\mathbb{N}\big\rangle\!\big\rangle ...
8
votes
5answers
428 views

A confusion about Axiom of Choice and existence of maximal ideals.

The proof of the theorem of statement that every ring has a maximal ideal uses zorn's lemma or the axiom of choice.Now, the defintion of ring as well as the definition of maximal ideal don't depend on ...
2
votes
1answer
69 views

Superfluous assumption in a counterexample to Frobenius algebras

In the wikipedia entry on Frobenius algebras, there are some examples and counter-examples. In example 5, where do you need that $\operatorname{char}(k) \neq 2$ ? I think $R:= k[x,y]/ (x,y)^2$ is ...
2
votes
2answers
127 views

Radicals of subrings

It is known that for a subring $R$ of some (commutative) ring $S$, the nilradical of $R$ $$\text{nil }R=R\cap\text{nil }S.$$ Moreover for Jacobson rings $R\subset S$, this means that the Jacobson ...
5
votes
3answers
406 views

Non-UFD integral domain such that prime is equivalent to irreducible?

In the integral domain every prime is irreducible. But the converse is not true, for example, $1+\sqrt{-3}$ is an irreducible but not a prime in ${\Bbb Z}[\sqrt{-3}]$. In a UFD, "prime" and ...
5
votes
2answers
353 views

If $R$ is an infinite ring, then $R$ has either infinitely many zero divisors, or no zero divisors

Please help me to prove that if $R$ is an infinite ring, then $R$ has either an infinite number of zero divisors, or it has no zero divisors.
1
vote
3answers
422 views

How to find the inverse of a polynomial in Laurent series division ring?

In the Laurent series division ring, how can we find the inverse of a given polynomial, for example: $3x^{-2} +x^{-1}+5x+7x^4$. Is there a certain formula to find the inverse? I've tried to find the ...
2
votes
0answers
50 views

Left ideals of central simple algebra generated by symmetric element

Let $F$ be a field of characteristic $\neq 2$. Suppose $(A,\sigma)$ is central simple algebra over a field $F$ with an orthogonal involution. Assume $(A,\sigma)$ is non-split. How to show that every ...
1
vote
3answers
94 views

$\phi$ in $O_K$ but not in $\mathbb{Z}[t]$

I have this problem: Let $t$ be a root of the polynomial $f(x) = x³ + x² - 2x + 8$. Let $\phi = \displaystyle \frac{4}{t}$ and let $K = \mathbb{Q}(t)$. I was able to show that $f(x)$ is irreducible, ...
9
votes
3answers
1k views

How to show that a finite commutative ring without zero divisors is a field?

$R$ is finite commutative ring without zero divisors which has at least two elements. How to show that $R$ is field? I'm just starting with abstract algebra and I'd really appreciate if someone ...
5
votes
3answers
1k views

Maximal ideals in polynomial rings

Let $K$ be a field. Let $\mathfrak{m}$ be an ideal of the polynomial ring $K[x_1,\ldots,x_n]$ and suppose the quotient $\frac{K[x_1,\ldots,x_n]}{\mathfrak{m}}$ to be isomorphic to $K$ itself. I want ...
3
votes
4answers
450 views

Understanding the quotient ring $\mathbb{R}[x]/(x^3)$.

I am having difficulty in understanding exactly the elements of the set $\mathbb{R}[x]/(x^3)$. I'll explain my thought process. The Quotient Ring is the set of additive cosets, so we have that ...
1
vote
1answer
170 views

Properties inherited from $R$ by Laurent polynomials $R[x;x^{-1}]$

I wonder if there is a paper about the conditions going up to Laurent Polynomial rings For example the Laurent polynomial preserves the condition of reversibility of ring R For a ring $R$ ...
4
votes
3answers
569 views

Ring of all continuous functions from reals into reals is not integral domain

Let $R$ be the ring of all continuous functions from the real numbers into the real numbers. Prove that $R$ is not an integral domain. I need help with this. I do not understand this at all and my ...
6
votes
3answers
398 views

Does Euclid lemma hold for GCD domains?

Exercise $ 10 $ of Section $ 3 $ of Chapter III of Hungerford’s Algebra states that if $ R $ is a UFD and if $ a,b $ are relatively prime, then $ a | bc $ implies $ a | c $, something that is easy to ...
1
vote
1answer
52 views

Conjugacy class in matrix ring

Let $M_{2}(\mathbb{R})$ be the ring of $2\times 2$ matrices over the reals and $M_{2}(\mathbb{R})^*$ the set of invertible such matrices. Consider any $A \in M_{2}(\mathbb{R})$ such that $ A^{2}=-I$, ...
1
vote
1answer
61 views

$P$ is prime if and only if for every pair of ideals $I,J$ containing $P$ properly, we have $IJ\nsubseteq P$

I'm trying to show that if $P$ is an ideal of a ring $A$; $A$ not necessarly conmutative, then $P$ is prime if and only if for every pair of ideals $I,J$ containing $P$ properly, we have ...
7
votes
1answer
303 views

Computing the ring of integers of a number field

This question arose from the need to see the splitting behaviour of primes over an extension of number fields. One criterion is the Kummer's theorem, which gives the decomposition of the base prime in ...
2
votes
2answers
71 views

Why is $S/R$ a ring extension?

If $S$ is a ring and $R \subset S$ is a subring it's common to write that $S/R$ is an extension of rings. I frequently find myself writing this and read it quite often in textbooks and lecture notes. ...
1
vote
1answer
49 views

Proof of a linear transformation property

Suppose $\phi:X \rightarrow Y$ is a map of sets and $F$ is a field. Let $\phi^* : F(Y) \rightarrow F(X)$ be a map sending a function $f \in F(Y)$ to a function $\phi^*(f) \in F(X)$ given by ...
9
votes
1answer
233 views

If $I$ is a finitely generated ideal of $A[X]$, is $I\cap A$ necessarily finitely generated for a commutative unital ring $A$?

Let $A$ be a commutative ring with $1$ and $A[X]$ the ring of polynomials in one variable over $A$. Assume $I$ is a finitely generated ideal of $A[X]$. My question is Is $I\cap A$ necessarily ...