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
40 views

Prove that $C(X,\mathbb R)$ has no non trivial nilpotent elements.

Can anybody help me in this question. I have no idea how to proceed. Any HINT will be appreciated.
1
vote
1answer
25 views

Show that the map is bijective.

I have a doubt in part $(3)$ Clearly this map is surjective using part $(b).$ To show injectiveness, let $x\neq y$ To show: $M_x\neq M_y$ Now as we are working in a Hausdroff space so ...
-4
votes
1answer
28 views

Quotient ring $(\mathbb{Z}_4 \times \mathbb{Z}_6)/S$

Consider the ring Z4xZ6 with +6, *6, and +4, *4 in appropriate coordinates and S={(0,0),(2,0),(0,3),(2,3)}. Would the elements of the quotient ring Z4 x Z6 / S be: S+0 (trivial set above), ...
1
vote
1answer
28 views

Modules over rings which are NOT a PID, or NOT a UFD [closed]

I am interested in studying the properties of modules over rings which are not Principal Ideal Domains or are not Unique Factorization Domains, but I am finding it very difficult to find any material ...
0
votes
0answers
11 views

Linear independence in a module

It is widely known that for any matrix on a commutative field, the following properties are equivalent : 1. Determinant is invertible 2. Matrix has an inverse 3. The only zero linear combinations ...
0
votes
1answer
32 views

Does the group $G$ of $n$th roots of unity form a subring of $\Bbb C$?

Is it true that the group $G$ of $n$th roots of unity is a subring of $\Bbb C$? My initial thought is that this is most definitely not true because the element $0$ is not an $n$th root of unity, and ...
-4
votes
2answers
50 views

Set theory with abstract algebra [closed]

Let R be an integral domain and let $x,y \in R$ nonzero. Prove that $xR = yR$ if and only if $y = xu$ for some unit $u$. Been having a lot of trouble with this. I'm not very good at this abstract ...
1
vote
0answers
34 views

Polynomial roots in the ring extension

Let $R$ be a ring with identity (not necessarily commutative) and $R[x]$ be a ring of polynomials over $R$. We say that a ring $S$ is an extension of $R$ if there is a subring $\tilde{R}$ in $S$ ...
2
votes
1answer
17 views

Upper Nilradical of a Ring

If we define the upper nilradical of a ring as the sum of all nil ideals of the ring, how could we deduce from just this definition that this is a nil ideal? Thanks!
0
votes
0answers
18 views

What is the difference between ${\mathbb{Z}[x_1, .., x_n]}_{( p )}$ and ${\mathbb{Z}[x_1, .., x_n]}/(p)$?

Let $p$ be any prime. What is the difference between ${\mathbb{Z}[x_1, .., x_n]}_{( p )}$ and ${\mathbb{Z}[x_1, .., x_n]}/(p)$? ${\mathbb{Z}[x_1, .., x_n]}_{( p )}$ is the localization of ...
1
vote
1answer
15 views

Extending Semiring $\mathbb{N}$ to $\mathbb{Z}$ through exact sequence

I am working on extensions in the form of $$A\hookrightarrow B\twoheadrightarrow C$$ in my thesis and I am just wanting to add as an extra note, IF POSSIBLE, this. We have that that $\mathbb{Z}$ is ...
1
vote
1answer
32 views

Let $M$ be the maximal ideal in $C(X,\mathbb R)$. Prove that there exists $x\in X$ such that $M=M_x$. [duplicate]

I have done part $(a)$ by defining a map from $C(X,\mathbb R) \to \mathbb R $ as $\phi (f)=f(x) $ and got the $M_x$ as kernel of homomorphism and got the answer. But I am unable to solve for ...
9
votes
3answers
79 views

Can we recover a compact smooth manifold from its ring of smooth functions?

It is well-known that if $X$ is a reasonably nice topological space (compact Hausdorff, say) then we can recover $X$ from the ring $C(X)$ of continuous functions $X\to\mathbb R$; see this MSE question ...
2
votes
0answers
22 views

For rings: containment and isomorphism does not imply equality.

So I have been asked to find two commutative unital rings $A$ and $B$ such that $B\subseteq A$ and $A\cong B$ but $A\neq B.$ I give my solution below. I would be very grateful if someone could ...
2
votes
3answers
41 views

$\mathbb{Z} [\sqrt{2}]$ is an integral domain

We know that $(\mathbb{Z} [\sqrt{2}],+,\cdot)$ is an integral domain. Someone can prove it easily if he says that is a subring of $(\mathbb{R} ,+,\cdot)$ . Can we find a different proof, more ...
1
vote
4answers
33 views

Isomorphism of a ring of matrices

Is it possible for a ring of matrices to isomorphic a ring of numbers? Suppose $$R = \begin{pmatrix} a & b \\ -3b & a \\ \end{pmatrix} a,b \in \mathbb Z $$ Can $R$ ...
2
votes
2answers
48 views

Are Z and Z* (defined below) isomorphic as rings?

Define $\mathbb{Z}^*$ to be the set of integers but with the following operations: $a \circ b = a + b - 1$ and $a * b = a + b - ab$ where $a+b$ and $ab$ are the usual integer addition and ...
0
votes
0answers
22 views

Is there any proof of the criterion of determining maximal ideal in a commutative ring with unity by Third Isomorphism Theorem?

Theorem. Let $R$ be a commutative ring with unity $1$ and $M$ is an ideal of $R$. Show that $M$ is maximal iff $\dfrac{R}{M}$ is a field. In the proof of this theorem the methods so far I have ...
1
vote
1answer
44 views

Integral extension and s.o.p.

Let $R\subset S$ be an integral extension. Is a system of parameters of $R$ a system of parameters of $S$ and conversely? I think so, since there is good behavior in dimensions. Many thanks.
7
votes
1answer
45 views
+50

On those integers $n>1$ such that any commutative ring with identity having exactly $n$ ideals is a PIR

Convention : All rings are commutative with unity unless stated otherwise. By ideals we will mean to include $\{0\}$ and $R$ also. Let us call an integer $n>1$ a "principal number" if any ring ...
0
votes
2answers
27 views

On those integers $n>1$ such that there exists a commutative ring with identity with exactly $n$ ideals

Let $n>1$ be an integer; we call $n$ a "ring number" if there exists a commutative ring $R$, with identity, having exactly $n$ ideals (including $\{0\}$ and $R$); now since for every $n>1$, ...
0
votes
0answers
18 views

$\mathbb{Z}[\frac{1-\sqrt{-19}}{2}]$ - principal ring, but not an euclidean ring [duplicate]

I am stucked on the problem. Is there someone who could tell me why $\mathbb{Z}[\frac{1-\sqrt{-19}}{2}]$ is a principal ring, but it is not an euclidean ring?
-1
votes
1answer
31 views

Factoring polynomials in $\Bbb Z_n$

a). Factor $f(x) = x^3 + 4x^2 + 5x + 2$ completely over $\Bbb Z_7$. b). Give two different factorizations of $x^2 + x + 8$ in $\Bbb Z_{10}[x]$. I have found the zeros of both of these but I am ...
0
votes
1answer
57 views

Find all the zeros of $f(x) = x^3 + 3x + 5$ in $\Bbb Z_7$

Find all the zeros of $f(x) = x^3 + 3x + 5$ in $\Bbb Z_7$. I've tried factoring this into multiple forms but I can't seem to find an easy way to find the $x'$s for $x^3 + 3x + 5 = 0$. Any hints or ...
0
votes
1answer
21 views

Two questions regarding polynomial rings.

Give an example of a natural number $n > 1$ and a polynomial $f(x) ∈ \Bbb Z_n[x]$ of degree $> 0$ that is a unit in $\Bbb Z_n[x]$. For this is set $n=2$. So then $f(x) = x \in \Bbb Z_2[x] $. ...
0
votes
2answers
47 views

General questions about Polynomial Rings [closed]

I'm learning about polynomial rings in my class. My instructor and book are both spectacularly unhelpful and didn't even bother to define most of the terms in my homework. So I have some general ...
1
vote
1answer
26 views

Let R* be the set of units of R and S* be the set of units of S. Prove that f(R*) = S*.

Let R and S be commutative rings with unity $1_R$ and $1_S$ respectively, and let $f: R\to S$ be a ring isomorphism. I am at a loss. Any help is much appreciated.
0
votes
2answers
31 views

$f(x) \in \mathbb{R}[x]$. Prove that if $z = a + bi$ is a zero of $f(x)$ then $z = a − bi$ is also a zero of $f(x)$.

Let $f(x) \in \mathbb{R}[x]$. Prove that if $z = a + bi$ is a zero of $f(x)$ then $z = a − bi$ is also a zero of $f(x)$. I'm learning about polynomial rings but my book and my instructor never ...
6
votes
1answer
99 views
+50

$R$ be a commutative ring with unity such that every surjective ring homomorphism $f:R\to R$ is injective , then is $R$ Noetherian?

Let $R$ be a commutative ring with unity such that every surjective ring homomorphism $f:R\to R$ is injective , then is $R$ a Noetherian ring ?
-1
votes
0answers
38 views

$A,B\in\mathbb Q[x]$ with $A,B$ monic, and $ AB\in\mathbb Z[x]$, prove $A,B\in\mathbb Z[x]$

It is part of cyclotomic polynomials. But I don't know how to deal with it and what to do next. I have prove $n$-th root is related to Euler's totient fuction. But I don't know how to use it. Thank ...
9
votes
1answer
81 views

Commutativity of a ring from idempotents.

In a ring $R$ with unity, every element can be written as product of finitely many idempotents. Can one show that the ring is commutative?
-4
votes
1answer
28 views

The mapping defines a unique automorphism [closed]

Let $R$ be a commutative ring with unity and $a,b\in R$ with $a$ invertible. I want to show that the mapping $x\rightarrow ax+b$ defines a unique automorphism of $R[x]$ that is identity in $R$. ...
1
vote
1answer
63 views

Do we have to show that $f(x)\in R$?

Let $R$ be a commutative ring with unity. I want to show that if $g(x)=c_nx^n+\dots+c_0\in R[x]$ is a zero divisor of $R[x]$ then there exists $d\in R \setminus \{0\}$ such that $dc_n=dc_{n-1}=\dots ...
1
vote
2answers
27 views

Why is $\varphi(X_i) = X_i + b_i$ an automorphism of $K[X_1,\dots,X_n]$?

I'm trying to justify to myself the assertion (used here) that given a field $K$ and elements $b_1,\dots,b_n\in K$, the map $\varphi(X_i) = X_i + b_i$ is a $K$-automorphism of $K[X_1,\dots,X_n]$. ...
3
votes
1answer
49 views

Rings in which $ab=0$ implies $axb=0$

I'm sure there must be some standard term for (not necessarily commutative) rings $R$ in which $ab=0$ implies $(\forall x)\, axb=0$ (for example, this is the case if $R$ is commutative or is a ...
0
votes
2answers
29 views

$\mathbb F_q[x]/(p(x))$ is a field of order $q^n$.

Let $\mathbb F_q$ be a field of order $q$ and $p(x)$ be an irreducible element in $\mathbb F_q$ of degree $n$. Then prove that $\mathbb F_q[x]/(p(x))$ is a field of order $q^n$. Attempt: As $p(x)$ ...
2
votes
1answer
44 views

Each proper ideal is a product of prime ideals

$R$ is a commutative ring with unity. If $R$ is P.I.D. I want to show that each of its proper ideal is written as a product of prime ideals. $$$$ Since $R$ is a P.I.D. every ideal is a prime ...
2
votes
1answer
32 views

$R$ be a finite commutative ring such that distinct ideals of $R$ have distinct orders (size) ; is $R$ a PIR?

Let $R$ be a finite commutative ring such that distinct ideals of $R$ have distinct orders (size) ; then is $R$ a Principal ideal ring (PIR) ? What if we moreover assume that distinct subrings of $R$ ...
2
votes
1answer
55 views

Is there an adjective for rings whose every non-zero prime ideal is maximal?

(All my rings are commutative and unital.) Question. Is there an adjective for rings whose every non-zero prime ideal is maximal? Remarks: Every PID has this property; more generally, every ...
1
vote
2answers
33 views

Let $M$ be a non zero maximal ideal in $\mathbb C[x].$ Prove that there exists $a\in \mathbb C$ such that $M=\langle x-a\rangle$.

Let $M$ be a non zero maximal ideal in $\mathbb C[x].$ Prove that there exists $a\in \mathbb C$ such that $M=\langle x-a\rangle$ (ideal generated by $x-a$). Attempt: As $\mathbb C[x]$ is a PID, ...
0
votes
0answers
21 views

If $R$ is a ring and $M$ is a left simple $R$-module, then $R/ann_{R}M$ is a left primitive ring

I'm attempting to prove that if $R$ is a ring and $M$ is a left simple $R$-module, then $R_1=R/ann_{R}M$ is a left primitive ring. I know that this becomes trivial if M is a faithful simple left ...
1
vote
3answers
28 views

Annihilator of modules [duplicate]

If $A$ is an $R$-module, I am having difficulty proving that $A$ is also a well-defined $R/ann(A)$-module with $(r+ann(A))a=ra$.
2
votes
0answers
32 views

What is the intuition behind a Euclidean function?

Many algebra textbooks give the definition of a Euclidean domain as an integral domain $R$ equipped with a Euclidean function/map (let's call it $\nu$). What I don't understand is the significance of ...
1
vote
1answer
34 views

Book recommendation on Primary decomposition of ideals [closed]

I'm trying to prepare a presentation on "Primary Decomposition of Ideals" which is the title of my project. But I'm new for the subject so I need help on the following points How to outline my ...
-5
votes
1answer
39 views

In a ring $x^2= 0$ implies $x=0$. Then every idempotent is central. [closed]

In a ring $x^2= 0$ implies $x=0$. Then every idempotent is central.
0
votes
0answers
11 views

The module of infinite matrices has bases with any length, isn't it? [duplicate]

Let $R$ be a ring. We consider matrices of elements from $R$ with the following properties: The sizes of a matrix is infinite; Any row of a matrix have a finite number of nonzero elements of $R$ ...
1
vote
1answer
54 views

Associativity of the tensor product of bimodules

Let $A_0,\dots,A_n$ be algebras over some fixed commutative ring $k$ (you may assume $k=\mathbb{Z}$ for simplicity). Let $M_i$ be an $(A_{i-1},A_i)$-bimodule for $i=1,\dots,n$. A multilinear map from ...
2
votes
1answer
23 views

Direct Summands of PI Rings as Right Ideals

Is any direct summand of a PI-ring (polynomial identity ring) necessarily idempotent as a right ideal? The answer is yes for a special case of PI-rings, namely any direct summand of a ...
0
votes
1answer
45 views

Finding the maximal ideals of the quotient of a polynomial ring by an ideal

Let the field $k$ be algebraically closed, let $f(X) \in k[X]$ be a separable polynomial of degree at least $2$, let $$ B = \frac{k[Y,X]}{(Y^2 - f(X))} $$ and write $y,x$ for the images in $B$ of $Y$ ...
0
votes
0answers
20 views

Ring $R=\{a+b\sqrt{-5} | a,b\epsilon Z $} [duplicate]

Show that in the Ring $R=\{a+b\sqrt{-5} | a,b\epsilon Z $} the element $\alpha$= 3 and $\beta= 1+2\sqrt(-5)$ are relatively prime, but $\alpha \gamma$ and $\beta\gamma$ have no GCD in R, where ...