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
votes
1answer
100 views

Ring homomorphism - isomorphic subring

Please help me with this exercise. Let $K$ be a field, $A$ a ring (without the assumption that a ring must have an identity) and $\phi : K \longrightarrow A$ an homomorphism. Prove either $A$ must ...
3
votes
1answer
265 views

Proof about Noetherian rings

I have to prove that every finite ring is Noetherian. I know examples of Noetherian rings which are not finite such as the field of complex numbers or a PIR like the integers. But anyway: [Proof]: I ...
3
votes
3answers
116 views

If $M$ is an $R$-module and $I\subseteq\mathrm{Ann}(M)$, then $M$ has the structure of an $R/I$-module

Let $M$ be an $R$-module and let $I$ be an ideal of $R$ such that $I$ is a subset of $\mathrm{Ann}(M)$. Define a product of an element of $R/I$ by an element of $M$ as follows: ...
2
votes
1answer
171 views

Reduced Gröbner Basis for $u:=X^2+2XYZ$ and $v:=XY+2Y^2Z-1$

I have $u:=X^2+2XYZ$ and $v:=XY+2Y^2Z-1$ with lex $X>Y>Z $. I have calculated the Gröbner Basis as $G=\{ X^2+2XYZ, XY+2Y^2Z-1, X, 2Y^2Z-1 \}$. But the question I have asks for the Reduced ...
5
votes
2answers
334 views

Equivalent definitions for projective modules

Fact: Let $R$ be a ring with identity. Let $J$ be an $R$-module. Then, $J$ is injective iff for every left ideal $L$ of $R$ every $R$-module homomorphism $L\rightarrow J$ can be extended to an ...
1
vote
2answers
62 views

Integral extensions

Let $p\neq1$ be an integer and let $\beta$ be a root of $x^6-p$. What is the difference, in terms of $\mathbb{Z}$-modules, between $\mathbb{Z}[\beta]$ and $\mathbb{Z}[\beta^2,\beta^3]$? I can ...
1
vote
1answer
26 views

Why $e_1A=M_1$?

Let A be a ring with identity $1$ and $M_1, M_2$ submodules of $A$. We have $1=e_1+e_2$, where $e_i\in M_i$, $i=1, 2$. We can show that $e_i$ are idempotent and $e_1e_2 = e_2e_1 = 0$. We have ...
2
votes
2answers
87 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
364 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
160 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
720 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
46 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
35 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
53 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
97 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 ...
7
votes
3answers
392 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
121 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 $p$ ...
3
votes
1answer
155 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
86 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
240 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
204 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
126 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
240 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 ...
4
votes
1answer
250 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
164 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
446 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
902 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}] ...
5
votes
1answer
163 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 ...
-1
votes
1answer
150 views

How to remove intersection of ideal $I$ and $J$ from union of ideal $I$ and $J$

after get the intersection of ideal $I$ and ideal $J$ how to remove this intersection from union of ideal $I$ and ideal $J$ in order to do prime decomposition how can it do in maple? actually i ...
9
votes
4answers
659 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
247 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
92 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. ...
5
votes
3answers
442 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
969 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
179 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
129 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 + $\forall x(Sx=x+1)$ + ...
6
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
148 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
470 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
136 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
455 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 ...
7
votes
2answers
414 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
480 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
96 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
2k 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 ...
6
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
501 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
176 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 $R$ ...