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 (1)

1
vote
1answer
41 views

relation between units and non zero divisors in a ring

I can prove that in finite commutative ring, non zero divisors are units. My question is if the reverse also true. I mean, units are non zero divisors? And what about the commutative infinite rings?
3
votes
1answer
20 views

$f:R \to D$ a homomorphism of the additive group of rings , $f(aba)=f(a)f(b)f(a) , f(1_R)=1_D$ , then is $f$ a ring homomorphism?

Let $R$ be a ring with multiplicative identity $1$ and $D$ be an integral domain with multiplicative identity ( i.e. $D$ is a commutative unital ring without zero divisors ) , let $f:R \to D$ be a ...
3
votes
2answers
59 views

Can $ℂ$ be viewed as a (nontrivial) field of fractions?

Is there an interesting ring $S ⊂ ℂ$ such that $ℂ = Q(S)$? I’m thinking no, but how can I prove it?
-1
votes
0answers
63 views

Find the number of elements of quotient rings

Let $R$ be the ring obtained by taking the quotient ring of $\mathbb Z_6[x]$ by the principal ideal $(2x + 4)$. Then $R$ has infinitely many elements. we know that $2x + 4 = 0$ $\Rightarrow$ $ x ...
2
votes
0answers
62 views

Localization of euclidean ring is euclidean?

I am trying to prove that a localization of a euclidean ring is euclidean, and the converse statement. I feel the basic definition of the norm is enough but I do not know how. Please note I am very ...
4
votes
2answers
48 views

Question about kernel and homomorphism

I was wondering is there any reason we take the identity e` for the kernel for ring homomorphism to be the additive identity instead of the multiplicative one?
0
votes
1answer
50 views

Nakayama's lemma, second version

Let $R$ be a commutative ring with identity, $J$ an ideal that is contained in every maximal ideal of $R$, and $A$ is finitely generated $R-$ module. If $R/J\otimes _R A=0$, then $A=0$. ...
1
vote
0answers
27 views

Noetherian local ring with maximal ideal $M$

Let $R$ be a Noetherian local ring with maximal ideal $M$. If the ideal $M/M^2$ in $R/M^2$ is generated by $\{ a_1+M^2, \dots, a_n +M^2\}$, then the ideal $M$ is generated in $R$ by $\{ a_1, \dots , ...
0
votes
2answers
32 views

Clarification on quadratic ring notation

My Abstract Algebra text is using the notation $\mathbb{Z}[1 + \sqrt{-5}]$ and calling it a "quadratic integer ring." Just to clarify, $\mathbb{Z}[1 + \sqrt{-5}]$ is simply the set $$ \left\{ a + b(1 ...
0
votes
2answers
47 views

Confused on notions of maximal ideal and some notation

I'm just getting started learning ring theory and am currently learning about ideals. By book (Dummit & Foote) says the following: For example, in the ring $R = \mathbb{Z}[x]$ the elements $2$ ...
2
votes
1answer
22 views

Isomorphism between modules over a semisimple ring

If $P$ is a module over the semisimple ring $R/J$, where $R$ is a semilocal ring having $1$, and $J$ is its Jacobson radical, does any isomorphism $P⊕...⊕P≅P'⊕...⊕P'$ with the same (finite) number of ...
0
votes
0answers
14 views

From progenerators to progenerators

I know that if $R$ is a ring (with identity) and $P$ is a progenarator right $R$-module (a f.g. projective generator) then $P/PJ$ is clearly f.g. when $J$ is the Jacobson radical of $R$; but, how ...
5
votes
1answer
91 views

Rings where every subgroup of the additive group is an ideal?

The rings $\mathbb Z$ and $\mathbb Z/n\mathbb Z$ have the property that every subgroup of the additive group is also an ideal (i.e., every subgroup absorbs multiplication by all ring elements). This ...
1
vote
1answer
32 views

one to one correspondence of Ideals in a ring and its localization

Let $A$ be a commutative ring, and $S$ a mutiplicatively closed subset. In my text book, it is stated that: there is one to one correspondence of prime ideals in ring $A$ (not meeting $S$) and ...
2
votes
0answers
31 views

A question about a system of congruent equations? Is there a unified proof by using ring theory?

In his book ``Topics in number theory, Volumes I and II''. William J. Leveque proved the following theorem(see page 34) Theorem A necessary and sufficient condition that the system of congruences ...
4
votes
2answers
71 views

Ring whose all ideals are double-sided is commutative?

I was thinking about the following problem: Suppose R is a ring s.t. every left ideal is also right. Is R commutative? This actually continues the easier question: Suppose G is a group whose ...
0
votes
1answer
42 views

Intersection of distinct maximal ideals in a commutative ring with identity.

If $R$ is a commutative ring with identity and $M_1, \dots, M_r$ are distinct maximal ideals in $R$, then show that $M_1\cap M_2 \cap \cdots \cap M_r = M_1M_2\cdots M_r$. Is this true if "maximal" is ...
0
votes
0answers
23 views

$a=3X^2+X+2 \in \mathbb{Z}_7[X]$. Compute the inverse of $[a]$ in $\mathbb{Z}_7[X]/(X^3+4)$

$a=3X^2+X+2 \in \mathbb{Z}_7[X]$. Compute the inverse of $[a]$ in $\mathbb{Z}_7[X]/p(x)$ where $p(x)=X^3+4$ I know that we want some $[b]\in\mathbb{Z}_7[X]/p(x)$ such that $[a][b]=[1]$. I used ...
0
votes
1answer
55 views

If a=b, how does a-b=0?

Let R be a ring and let $a,b \in R$. Suppose $a=b$. How do you know that $a +- b=0$? In other words, what axiom or step lets you say that $a +-b = b + -b$? (That you can add the same thing to both ...
0
votes
0answers
61 views
+50

Two properties related to semisimple rings

Let $R$ be a semisimple ring Show the following (i) If $xy=1 \in R$, then $yx=1$. (ii) If $x \in R$ is such that $xR$ is a left ideal of $R$, then $xR=Rx$. I am pretty lost with the two items. I ...
1
vote
1answer
42 views

Finite Extension of Integral Domains.

Let $D\subset E$ (integral domains), with fraction fields $k\subset K $. Suppose that $E$ is integral over $D$, and $E$ is $D$-module finitely generated. My question is: $[K:k]$ is finite? Thank ...
5
votes
2answers
362 views

Is there an easy way to remember the ring axioms?

I'm working on a problem that asks me to prove a set is a ring. I have to look up the axioms to prove it... I was curious, is there an easy way to remember the ring axioms, so that you can avoid ...
2
votes
1answer
27 views

Maximal and prime ideals of $2 \mathbb Z$

What are the all maximal ideals of $2 \mathbb Z$ ? what are the all prime ideals of $2 \mathbb Z$ ? We know that if $R$ is a commutative ring with multiplicative identity $1$ and $M$ is a maximal ...
0
votes
1answer
15 views

$m,n>1$ are relatively prime integers , then are there at-least four idempotent (w.r.t. multiplication) elements in $\mathbb Z_{mn}$ ?

If $m,n>1$ are integers such that $g.c.d.(m,n)=1$ then is it true that there are at-least four elements in $\mathbb Z_{mn}$ such that $x^2=x$ ( i.e. idempotent ) ?
1
vote
1answer
24 views

If $x$ is an integer and $m$ is an element from a ring prove that $x(-m) = -(xm)$

So my approach to this was to break it into 3 cases. Where x is =,>,< 0. The cases where x = 0 and x > 0 are easy. But I'm struggling with x < 0. Here is what I have so far: Let $x = -y$ ...
2
votes
1answer
45 views

Looking for example of a commutative non-unital ring in which every maximal ideal is a prime ideal

Give example of a commutative non-unital ring in which every maximal ideal is a prime ideal. The motivation for this question is : It is known that if $R$ is a commutative ring with identity $1 \ne ...
0
votes
1answer
46 views

Application of Chinese remainder theorem

from Chinese remainder theorem, we know that of $m,n \in \mathbf{Z}$, $(m,n) = 1$, then $Z_m \otimes Z_n \cong Z_{mm}$ as ring isomorphism, but how it related to the application that $\phi(nm) = ...
0
votes
2answers
30 views

Prove that an exact sequence splits

Let $0 \to r\mathbb{Z}_n \to \mathbb{Z}_n \to s\mathbb{Z}_n \to 0$ where n =rs an exact sequence of $\mathbb Z$ modules the how can I prove the sequence splits if and only if $(r,s)=1$ The only thing ...
1
vote
3answers
35 views

Let $f$ be a surjective homomorphism. Prove that $\ker(f)$ is a maximal ideal

Let $f:R\to S$ be a surjective homomorphism, where $R$ is a commutative ring and $S$ is a field. Prove that $\ker(f)$ is a maximal ideal. I already know that $\ker(f)$ is an ideal of $R$. I tried to ...
0
votes
1answer
34 views

How to determine degree of polynomials?

Let $A$ an integral domain and $f, g \in A[X]$ such that $\partial (f+g)=5$ and $\partial(f-g)=2$. Determine $\partial (fg)$; $\partial(f^{2}-g^{2})$ and $\partial(f^{2}+g^{2})$. P.S.: $\partial$ ...
1
vote
1answer
57 views

What ring is isomorphic to factor-ring?

$R=\mathbb{Z}[x]$ - polynomials with integer coefficients $I = (x^2+x-1)\mathbb{Z}[x]$ In this case we have that classes of factor-ring $R/I$ represented in the next form: $K_{a,b}=ax+b$. What ring ...
2
votes
1answer
31 views

Different forms for the exterior power of a module

First I have defined the exterior algebra of a module $M$ as the quotient $T(M)/A(M)$ where $T(M)$ is the tensor algebra of $M$ and $A(M)$ is the ideal generated by all elements of the form $m\otimes ...
3
votes
2answers
43 views

Proving equivalent versions of faithfully flatness.

I was reading a proof of the the following theorem from Matsumura (p.47) There was something confusing about $(3) \implies (2)$ and $(2) \implies (1)$. Question 1 Here, it says $M \not= ...
1
vote
1answer
49 views

Is there a general method to find if ideal is maximal

Is there an algorithm to determine if we have been given a ring $A$ and its ideal $I$, whether or not $I$ is a maximal ideal of $A$? I found that sometimes proving that ideal is maximal might be ...
1
vote
2answers
48 views

Example of a ring in which square of every element is zero

This is an exercise in a book "Rings and Modules- Musili": (in this book, ring may not have unity.) Give an example of a non-trivial commutative ring in which square of every element is zero. ...
2
votes
2answers
30 views

Factor each of the following elements in Z[i] into a product of irreducibles

1 + 3i Is there a way to find out if they are irreducible or by trial and error?
0
votes
2answers
26 views

How to show if it is irreducible

Let R be the subring (with identity) of Q[x, y] generated by x^2 , y^2 , and xy. Each of these elements is irreducible in R How do we know if they are irreducible
2
votes
2answers
32 views

Integral domain, UFD and PID related problem

(i) Let $R$ be an integral domain that has irreducible elements. Prove that $R[X]$ is not A PID. (ii) Let $R$ be a UFD and $K$ its field of fractions. Let $f \in R[X]$ be a monic polynomial ...
2
votes
1answer
60 views

Quotient of polynomial ring in two variables is a PID

With $K$ a field and $K[x,y]$ the polynomial ring over it in two variables, the quotient ring of it over the ideal generated by $1-xy$ is a PID. I've tried using Noetherianess but haven't gotten ...
0
votes
1answer
33 views

Definition of the subalgebra generated by a set

I can't seem to find a solid definition of the subalgebra generated by a set anywhere. Let $A$ be a commutative ring, $M$ an $A$-algebra and $S$ an $A$-submodule of $M$. We call $S$ a $subalgebra$ of ...
2
votes
1answer
41 views

Maximal ideal of the ring of all real valued continuous functions on $[0,1]$ [duplicate]

Let $R$ be the ring of all real valued continuous functions on $[0,1]$ and $M$ be a maximal ideal of $R$ ; then how to prove that $\exists c \in [0,1] $ such that $M:=\{f \in R:f(c)=0\}$ ?
3
votes
1answer
38 views

Let $R=M_n(D)$, $D$ is a division ring. Prove that every $R-$simple module is isomorphic to each other. [duplicate]

Let $R=M_n(D)$, $D$ is a division ring. Prove that every $R-$simple module is isomorphic to each other. I need some hints to prove it. Thank you very much.
4
votes
2answers
48 views

If there exists $n>1$ such that $x^n=x$ for all $x$ in a ring, then there are no nonzero nilpotent elements.

Suppose that there is an integer $n> 1$ such that $x^n=x$ for all elements $x$ of some ring. If $m$ is a positive integer and $a^m= 0$ for some a, show that $a = 0$. I have an answer but don't ...
-1
votes
1answer
67 views

Questions about the ring $R=(\Bbb Z/6\Bbb Z)[X]/(2X+4)$ [closed]

Let $R=(\Bbb Z/6\Bbb Z)[X]/(2X+4)$. Then A. $R$ has infinitely many elements? B. $R$ is a field? C. $5$ is a unit in $R$? D. $4$ is a unit in $R$? Can any one help me ..thanks for your time yes ...
1
vote
1answer
21 views

Finitely generated modules over a PIR (structure theorem application)

Let $R:=\mathbb R[x]/\langle (x^2+1)^2 \rangle$ and $J:=\langle x^2+1 \rangle \lhd R$. Prove that every finitely generated $R$-module is isomorphic to $R^m \oplus (R/J)^n$ for a unique pair of non ...
3
votes
1answer
92 views

Atiyah-MacDonald Ch. 4 exercise 20: what's the module analogue of $\sqrt{\mathfrak{a}+\mathfrak{b}} = \sqrt{\sqrt{\mathfrak{a}}+\sqrt{\mathfrak{b}}}$?

Atiyah-MacDonald exercises 20-23 in chapter 4 develop a theory of primary decomposition for modules, in analogy with the theory developed in the chapter for rings. Exercise 20 begins with this ...
1
vote
1answer
21 views

trace of left/right multiplication

Let $A$ be a finite dimensional algebra over some field $k$. Then any element $a \in A$ defines two $k$-linear maps $a_l, a_r$ on $A$ by left and right multiplication, respectively. So there are two ...
0
votes
2answers
22 views

Calculate the Kernel of $D$ with char $F=p$

Let $F$ be a ring and $f(x) = a_0 + a_1x + · · · + a_nx^n$ be in $F[x]$. Define $f'(x) = a_1 + 2a_2x + · · · + na_nx^n−1$ to be the derivative of $f(x)$. we can define a homomorphism of abelian ...
0
votes
1answer
20 views

Representatives of simple $\mathbb C[\mathbb D_3]$-modules (left modules)

Problem Let $\mathbb D_3$ be the symmetry group of the equilateral triangle. Give a complete list of the representatives of the simple left $\mathbb C[\mathbb D_3]$-modules. My attempt at a solution ...
1
vote
2answers
63 views

Are any of these rings isomorphic?

As part of my ongoing struggle to understand the complex conics, I've reached the following problem: Let $Q_1 = x^2 + y^2$, $Q_2 = x^2 - 1$, and $Q_3 = x^2$ be polynomials in $\mathbb{C}[x,y]$. ...