This tag is for questions about rings, which are a type of algebraic structure studied in abstract algebra and algebraic number theory.

learn more… | top users | synonyms (2)

3
votes
3answers
45 views

Is regular selection from recurrence also recurrence?

Let $R$ be a ring, and $u=(u(0),u(1),u(2),...)$ be a sequence over $R$ ($u(i)\in R$). Let $m\ge1$, $c_0,...,c_{m-1}\in R$ be fixed elements, and the following law of recursion holds $$u(i+m)=c_0u(i)+...
0
votes
2answers
37 views

Cancellation law of ideals in a certain ring

Let $R$ be a integral domain satisfying the following property. For any non-zero ideal $A$ of $R$, there exist $a \in R\ (a \neq 0)$ and a non-zero ideal $B$ of $R$ such that $AB=(a)$. ...
1
vote
3answers
41 views

How can I show the uniqueness of homomorphism?

Let $R$ be a commutative ring and let $k(x)$ be a fixed polynomial in $R[x]$. Prove that there exists a unique homomorphism $\varphi:R[x]\rightarrow R[x]$ such that $\varphi(r)=r\;\mathrm{for\; all\;...
2
votes
2answers
58 views

Fraleigh's proof that $ M $ is a maximal ideal if and only if $ R/M $ is a field

I was reading Fraleigh's abstract algebra textbook and he gave a proof about the theorem that if $ R $ is a commutative ring with unity, then $ M $ is a maximal ideal if and only if $ R/M $ is a field....
0
votes
0answers
10 views

The reduced norm map $\operatorname{Nrd}: K_1(A)\to K^\times$

Let $K_1(A)$ be the Grothendieck $K_1$-group of the category of finitely generated projective $A$-modules where $A$ is a central simple $K$-algebra. I'd be grateful if someone could tell me if the ...
0
votes
1answer
17 views

What is the reduced norm map?

This is a basic question about the reduced norm homomorphism. Let $A$ be a central simple $K$-algebra and $P$ a f.g. projective $A$-module. I know that $\operatorname{End}_A(P)$ is also a central ...
2
votes
2answers
71 views

$R$ commutative ring with unity , does polynomials with unit leading coefficients of degre s from $0$ to $n$ generate all polynomials of deg $\le n$?

Let $R$ be a commutative ring with unity , consider the polynomial ring $R[x]$ , let $\mathcal P_n:=\{f \in R[x] : f=0$ or $\deg f \le n\}$ , so $\mathcal P_n$ is a finitely generated module over $R$ ....
1
vote
1answer
24 views

We assume that there exists a ring homomorphism $f:k[x,y]/(\phi(x,y))\to k[t]/(t^2)$ that satisfy given conditions.

Let $k$ be a field, $r \in k$, and $\phi(x,y)=\sum a_{ij}x^iy^j\in k[x,y]$. We assume that there exists a ring homomorphism $$f:k[x,y]/(\phi(x,y))\to k[t]/(t^2)$$ satisfying: $f(a+(\phi(x,y)))=a+(t^2)$...
2
votes
2answers
53 views

To show that $\langle x-a , y-b\rangle$ is a maximal ideal of $F[x,y]$ by showing that $F[x,y]/\langle x-a , y-b\rangle$ is a field [duplicate]

Is there any way to show that for $a,b \in F$ , the ideal $\langle x-a , y-b\rangle$ is maximal in $ F[x,y]$ , by showing that the quotient $F[x,y]/\langle x-a , y-b\rangle$ is a field ? Is the ...
3
votes
1answer
52 views

$(x+a)^n=x^n+a$ in $\mathbb Z/n\mathbb Z[x]$ then n is prime.

Question If $n>1$ is an integer such that $(x+a)^n=x^n+a$ in $\mathbb Z/n\mathbb Z[x]$ for all $a \in \mathbb Z/n\mathbb Z$ then $n$ is prime. I want to show this with contradiction. But ...
1
vote
3answers
47 views

Is $M_a=\{f:f(a)=0\}$ a Principal ideal [duplicate]

Is $M_a=\{f\in \mathcal C[0,1]:f(a)=0\}$ where $\mathcal C[0,1]$ denotes the ring of continuous functions in $[0,1]$ a Principal ideal? My try: Let $M_a=\langle f_1\rangle$ .Consider $f(x)=\sqrt {|...
2
votes
1answer
35 views

Minimum cardinality module for a fixed finite ring

Let $F$ be a finite field and $k$ be a positive integer. Let $M_k(F)$ denote the ring of $k\times k$ matrices. $M_k(F)$ is an $M_k(F)$-module with matrix multiplication, and $F^k$ is an $M_k(F)$-...
0
votes
1answer
64 views

Every finite field can be obtained by a quotient of the ring $\Bbb Z[x]$.

Every finite field can be obtained by a quotient of the ring $\Bbb Z[x]$. My try: Any finite field $F$ is of the order $p^n$ where $p$ is a prime and $n\in \Bbb N$ . If we want to make a field of ...
3
votes
1answer
85 views

Isomorphism between endomorphism algebras

Assume that $R$ and $S$ are associative $\mathbb{C}$-algebras with unit $1_R$ and $1_S$, respectively. In addition, assume that $_RM$ is a simple left $R$-module and $_SN$ is a simple left $S$-module. ...
1
vote
1answer
59 views

A group-ring is commutative if and only if that group is abelian

Problem says: Let $R$ be a nontrivial commmutative ring and $G$ a group. Prove that $R[G]$ is commutative if and only if $G$ is abelian. I solved ($\Rightarrow $) direction as follow: ...
0
votes
0answers
22 views

Are there any integral domains in which irreducible elements are easily identified?

In every integral domain I've studied so far irreducible elements have been impossible to quickly identify in general with any known procedure. Is there an integral domain for which such a procedure ...
1
vote
0answers
46 views

$R_{a} = R[x]/(x)$ isomorphic to $R_{b} = R[x]/(x-1)$

I am looking at the following two rings: $R_{a} = R[x]/(x)$ and $R_{b} = R[x]/(x-1)$. I was told that these two rings were isomorphic, but I don't see why. Is this due to the minimal polynomials?
4
votes
1answer
48 views

Ideal of a product ring?

I am trying to prove whether or not the ideal generated by $\langle (2,2)\rangle$ is a prime ideal of $\mathbb Z_4\times \mathbb Z_4$? My issue is I'm not sure how to do the coordinate ...
1
vote
1answer
49 views

Prove $Q[x]/(x^2+4)$ is isomorphic to $Q[x]/(x^2+1)$

I've been asked to prove Q[x]/(x^2+4) is isomorphic to Q[x]/(x^2+1); I've looked at lots of similar solutions, but haven't been able to understand this. I know each ring is the quotient ring for their ...
0
votes
1answer
44 views

Characteristic of a product ring?

Let A and B be commutative rings with unity where char(A)=n and Char(B)=m s.t. n,m ∈ ℤ (and n,m ≠ 0). Prove or give counter example: if k ∈ ℤ+ and n,m both divide k, then Char (A x B)| k. Here was ...
0
votes
2answers
42 views

Intersection of two principal ideals is an ideal and lowest common multiple (if it is a PI)

I think the first part of the proof would go like this: any element in $(a) \cap (b)$ can be written as $ar_1 = br_2$, so multiplying by an element $r \in R$ yields $ar_1r\in aR$ or $br_2r \in bR$, so ...
3
votes
1answer
66 views

How to study for ring theory?

I want to study the theory of rings because it is used when I study representation theory. Here, a ring is not necessarily commutative and doesn't necessarily has unity. I know that there are a few ...
0
votes
2answers
23 views

Definition of ordered ring flawed?

Wikipedia says an ordered ring has these two properties: $a \leq b \rightarrow a+c \leq b+c$ $0 \leq a \land 0 \leq b \rightarrow 0 \leq ab$ Later the article says: $a \leq b \land 0 \leq c \...
0
votes
1answer
21 views

Reference on a result about integral closures.

Could you please give a reference or a sketch of a proof for the following proposition? Proposition: The integral closure of a complete local Noetherian domain $R$ is module-finite over $R$ You ...
-1
votes
1answer
26 views

Show that $[E:F] \le n!$ [duplicate]

Let $f(x)$ be a separable irreducible polynomial of degree n with coefficients in a field F. Let E be a splitting field of f(x) over F. Show that $[E:F] \le n!$
1
vote
2answers
38 views

Trying to show that $f$ is not a zero divisor?

Consider the ring $R=\dfrac {k[x,y,z,t]}{(y(xt-yz))}$. Consider the polynomial $f=t(y^3-x^2z)$. Is $f$ a non zero divisors of $R$? How do we check this? I know that if $f$ is a zero divisor then ...
0
votes
1answer
35 views

Is $f$ a non zero divisor?

Consider the ring $R=\dfrac {k[x,y,z,t]}{(y(xt-yz))}$. Consider the polynomial $f=t(yz-xt)$. Is $f$ a non zero divisor of $R$? I think the answer is yes, because if $f$ is zero divisor then $fh=y(yz-...
1
vote
3answers
56 views

Why does a ring homomorphism not necessarily map unit to unit?

I'm having trouble understanding why in a ring homomorphism, say maps from $R$ to $R'$, doesn't necessarily map the unit $1$ in $R$ to $1'$ in $R'$. If you use the definition that it preserves ...
0
votes
2answers
17 views

How would I work out the Cayley table for $F_3 [x]$ modulo $x^2 +2$ with addition and multiplication.

How would one display the Cayley table for $F_3 [x]/(x^2 +2)$ and show that it is a ring (I have assumed addition and multiplication are associative and that multiplication is distributive over ...
0
votes
1answer
22 views

The existence of a non-split composition series in a indecomposable module

Assume that $R$ is a ring with unit and $M$ is a indecomposable left $R$-module with finite length. That is, $M$ has a composition series. Is it true that there is a composition series $$\begin{...
3
votes
1answer
68 views

Prove that there exists a normal extension $F/\mathbb{Q}$ with $G(F/\mathbb{Q}) \cong\mathbb{Z}_{5}$.

Prove that there exists a normal extension $F/\mathbb{Q}$ with $G(F/\mathbb{Q}) \cong\mathbb{Z}_{5}$. I tried to solve this problem by thinking about a polynomial which has a splitting field of ...
0
votes
1answer
63 views

How many ideals in a ring R turned into Z/nZ

Say I have a ring R, is there any general way to find out how many ideals it has? I know that if it's a field then there are only 2 ideals, namely (0) and (1), however what if the ring is not a field, ...
2
votes
4answers
103 views

Prove that $\sqrt[3]{2} +\sqrt{5}$ is algebraic over $\mathbb Q$

Prove that $\sqrt[3]{2} +\sqrt{5}$ is algebraic over $\mathbb{Q}$ (by finding a nonzero polynomial $p(x)$ with coefficients in $\mathbb{Q}$ which has $\sqrt[3] 2+\sqrt 5$ as a root). I first tried ...
1
vote
1answer
21 views

If $R/P$ is an integral domain then $P\vartriangleleft R$ is prime. [duplicate]

Let $R$ be a ring and let $P$ be a proper ideal of $R$. If the quotient ring, $R/P$ is an integral domain then $P\vartriangleleft R$ is prime. For $x,y\in R$ we have $(x+P)(y+P)=xy+P\in P\...
0
votes
1answer
22 views

Fraction rings ideals members

Let $R$ be a ring with fraction ring $R_S$ and ideal $I$. I saw in arguments that when $a/s$ is in $I_S$ they dont say $a$ is in $I$. Instead they say $a/s=b/t$ with $b \in I$. Why? Many thanks.
0
votes
1answer
60 views

Rings where $ab=0$ for all elements

Let $R$ be a ring, not necessarily unital, such that $ab=0$ for all $a,b\in R$. Suppose $R$ only has trivial right ideals. Is it true that $R$ has finite order? Are these rings special?
2
votes
2answers
46 views

Transforming a Polynomial to Show Irreducibility Using Eisenstein's Criterion

I have a particular polynomial $$z^5-5z^4+30z^3-150z^2+465z-725$$ A quick check in mathematica shows that this polynomial is irreducible over the rationals, however, it does not pass the third ...
0
votes
1answer
32 views

Regarding taking powers of prime ideals in a ring

My question is simple to ask: given some prime ideal $P$ in a ring $R$, we can talk about $P^2, P^3$ etc. but can we discuss $P^0$? Is there a convention that says $P^0 = R$, or is there something ...
2
votes
0answers
29 views

What is the ring of integers in $\mathbb Q^c\otimes_K K_\mathfrak p$?

Let $K$ be a number field with ring of integers $\mathcal O_K$ and $\mathfrak p$ a prime of $K$. Let $\mathbb Q^c$ be the algebraic closure of $\mathbb Q$ in $\mathbb C$. If $L$ is a number field ...
0
votes
0answers
17 views

Describe the prime ideals of Ring $R$ in terms of their generators. [duplicate]

Let $R:=\Bbb C[x,y]$ denote the ring of polynomials in the variables $x$ and $y$, with complex coefficients. Describe the prime ideals of $R$ in terms of their generators. Prime ideals are ideals ...
0
votes
3answers
66 views

Show that two rings are not isomorphic [closed]

I don't know how to show (or why) $M_{2\times2}\mathbb{(R)}$ is not isomorphic to $\mathbb{R}[x]/(x^4-1)$ does it have something to do with the order of coset representatives of the quotient group?
0
votes
1answer
35 views

Define a new addition ⊕ and multiplication on Z by a⊕b = a + b−1 and ab = a + b−ab.

a+b and ab are the usual integer addition and multiplication. You can assume that this new operation forms a ring, say R is the set of integers with these operations. Then does R have zero-divisors? ...
0
votes
0answers
35 views

Determine the group of units of a subset of $M_n(\mathbb{C})$

Let $R$ be a commutative ring. Let $R=\bigg\{\begin{bmatrix}u & v\\ 0 & u\end{bmatrix}:u,v\in\mathbb{C}\bigg\}$. Determine the group of units $R^{\times}$ of $R$. My try: Let $A=\begin{...
0
votes
0answers
9 views

for every left ideal $L$ of $R$ and $R$-module homomorphism $g:L\to A$, there exists $a\in A$ such that $g(r)=ra$ for every $r\in R$.

Let be a $R$ ring with a identity. An $R$-module $A$ is injective iff for every left ideal $L$ of $R$ and $R$-module homomorphism $g:L\to A$, there exists $a\in A$ such that $g(r)=ra$ for every $r\...
1
vote
1answer
43 views

Do all fields have a total cyclic order?

It is well known the finite commutative rings, $Z/nZ$, are not discretely ordered rings. The axiom $\forall x \forall y \forall z((0<z \land x<y) \rightarrow (x*z < y*z))$ is false for the ...
3
votes
2answers
46 views

Multiplicative inverse of $x+f(x)$ in $\Bbb Q[x]/(f(x))$

So I have $f(x) = x^3-2$ and I have to find the multiplicative inverse of $x + f(x)$ in $\mathbb{Q}[x]/(f(x))$. I'm slightly confused as to how to represent $x + (f(x))$ in $\mathbb{Q}[x]/(f(x))$. ...
-3
votes
1answer
36 views

Kernel of a homomorphism: $f(a)=f(x)$? [closed]

Suppose $A$ and $K$ are rings with $f: A \to K$ a homomorphism. Prove that for any $x \in a + \ker(f)$ we have $f(x)=f(a)$. Im not sure how to start this, any help is appreciated!
0
votes
1answer
20 views

Ideal generated by given integers verification.

The question reads: Find the positive generator of the smallest ideal in $\mathbf Z$ containing the following ideals: a. $(4)$ and $(18)$. My answer is $(m)=(4)$. b. $(6)$ and $(35)$. My ...
-1
votes
2answers
32 views

if a is a unit of $A$, it is also a unit of the quotient ring? [closed]

Suppose $A$ is a nontrivial commutative ring with unity and $S$ is an ideal of $A$ s.t. $S\ne\{0_A\}$ and $S\ne A$ . Prove or find a counterexample: if $a\in A$ is a unit in $A$, then $a + S$ is a ...