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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6
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3answers
568 views

Does every prime ideal in a ring arise as kernel of a homomorphism into $\mathbb{Z}$?

Let $R$ be a commutative ring. Clearly the kernel of $h$ is a prime ideal whenever $h : R \rightarrow ...
4
votes
1answer
135 views

Why are projective modules contained in this class of modules?

Suppose $A$ noetherian and define $G(A):=\{M: M$ is an $A$-module reflexive and Ext$^i_A(M,A)=$Ext$^i_A(M^*,A)=0$ for $i\geq1\}$ Why are projective modules contained in this class? Of course if ...
1
vote
2answers
286 views

When are quotient maps induced by equivalence relations surjective and injective?

Let $\sim$ be an equivalence relation on a set $X$. Also, there is a natural function $p:X\to \tilde X$ where $\tilde X$ is a set of all equivalence classes. (Equivalence classes are defined as, ...
1
vote
2answers
1k views

proof of chinese remainder theorem for ring

Let $R$ be a ring(not necessary have "1") and let $I,J$ be ideals of $R$ such that $I+J=R$. I want to prove that there is a $x\in R$ such that $$x\equiv r ({\rm mod} I) \quad x \equiv s ({\rm mod} J) ...
3
votes
0answers
96 views

Applications of Govorov-Lazard Theorem?

The Govorov-Lazard Theorem states that a (right) module over an unital ring is flat iff it is a direct limit of finitely generated free (right) modules. I wonder if this theorem has interesting ...
13
votes
3answers
509 views

Pathologies in module theory

Linear algebra is a very well-behaved part of mathematics. Soon after you have mastered the basics you got a good feeling for what kind of statements should be true -- even if you are not familiar ...
5
votes
2answers
370 views

In a ring $aba=0$ implies $ab=0$ or $ba=0$?

In a ring, $a\neq0$ and $b\neq0$. $aba=0$. Prove $ab=0$ or $ba=0$. This is one question in my abstract algebra homework-- it seems pretty easy at first glance, yet I have spent hours thinking about ...
6
votes
2answers
526 views

Ring homomorphism

The number of non-trivial ring homomorphisms from $\mathbb{Z}_{12}$ to $\mathbb{Z}_{28}$ is (Options: ...
3
votes
1answer
126 views

Question about prime ideals and union of ideals

The questions asks to show that if $A$ is a ring and $I, J_{1}, J_{2}$ ideals of $A$, and $P$ is a prime ideal, then $I \subset J_{1} \cup J_{2} \cup P$ implies $I \subset J_{1}$ or $J_{2}$ or $P$. ...
2
votes
0answers
110 views

Multiplication structure for finite abelian rings of order $p^2$.

Consider an odd prime $p$ and a positive integer $a$ as close to 2 as possible that is not a quadratic residue $mod$ $p$. If we extend the ring $mod$ $p$ with the element $b = a^{\frac{1}{2}}$ then ...
1
vote
2answers
96 views

The Uniqueness of a Coset of $R[x]/\langle f\rangle$ where $f$ is a Polynomial of Degree $d$ in $R[x]$

Suppose $R$ is a field and $f$ is a polynomial of degree $d$ in $R[x]$. How do you show that each coset in $R[x]/\langle f\rangle$ may be represented by a unique polynomial of degree less than $d$? ...
0
votes
0answers
162 views

Factoring polynomials $f(g(x))$ over extension fields.

This question is a variation on another one : related question Let $f(x)$ and $g(x)$ be polynomials with integer coefficients and discriminant of $f(x)$ > discriminant of $g(x)$ , which are ...
0
votes
1answer
304 views

multiple choice question for compact support functions.

Let $C(\mathbb R)$ denote the ring of all continuous real-valued functions on $\mathbb R$, with the operations of pointwise addition and pointwise multiplication. Which of the following form an ideal ...
2
votes
1answer
174 views

What is wrong with this proof of Wedderburn's little theorem?

Wedderburn's little theorem $\quad$ every finite domain $A$ is a field. Proof $\quad$ Let $x$ be a nonzero element of $A$. Because $A$ is finite, there exist positive integers $n$, $k$ such that ...
0
votes
4answers
782 views

ring theory multiple choice question

Pick out the true statement(s): (a) The set of all $2\times 2$ matrices with rational entries (with the usual operations of matrix addition and matrix multiplication) is a ring which has no ...
2
votes
1answer
61 views

Some questions about rings

All rings are commutative and unital Q1: what means notation $$A\cong A_1\times\ldots\times A_n?$$ Is it true that elements of $A_1\times\ldots\times A_n$ are collection of elements of $A_1,\ldots ...
2
votes
1answer
133 views

a problem on ideal

Pick out the true statements: a. Let $R$ be a commutative ring with identity. Let $M$ be an ideal such that every element of $R$ not in $M$ is a unit. Then $R/M$ is a field. b. Let $R$ be as above ...
5
votes
2answers
106 views

For a ring $\{0,1,c\}$, does $c^2=1$?

Say you have an arbitrary ring with three elements, $\{0,1,c\}$. Why does it have to be that $c^2=1$? If we don't assume that $c$ is invertible, what goes wrong if $c^2=0$ or $c^2=c$?
5
votes
3answers
89 views

$(1-t)^{-d}= \Sigma_{k=0}^\infty {d+k-1 \choose d-1} t^k$?

I'm trying to see why the equation $(1-t)^{-d}= \Sigma_{k=0}^\infty {d+k-1 \choose d-1} t^k$ holds in the power series ring $\mathbb{Z}[[t]]$. I assume it's a counting argument about the number of ...
1
vote
1answer
123 views

Factoring polynomials of degree $a p^b$ over extension fields.

Let $f(x)$ be an irreducible polynomial with integer coefficients, which is irreducible over $\mathbb{Z}$ and has degree $a p^b > p$ with $p,a,b>0$ and $p$ a prime. It appears that $f(x)$ ...
2
votes
0answers
256 views

Field of Fractions for Commutative Ring with Identity

I'm trying to complete a problem that is walking me through creating a field of fractions $F$ from a commutative ring with identity (and no non-zero divisors) $R$. The construction first asks me to ...
2
votes
1answer
122 views

Equivalence classes on Z

So I'm given that R is an equivalence relation on Z, and that adding classes in the natural way is well defined. That is, class(a)+class(b) = class(a + b). I want to show that R can only be equality ...
1
vote
1answer
66 views

Notation $A // B$

Quick question: When $A$ and $B$ are Hopf algebras, what does $A /\!\!/ B$ stand for exactly? In context, it seems to be a type of quotient, but I need to be sure.
6
votes
1answer
242 views

polynomials over a local Artinian (or finite) ring

In this question " Zero-divisors and units in $\mathbb Z_4[x]$ " it looks like it has been shown that the set of zero divisors of $\mathbb{Z}_4[x]$ coincides with its nilpotent elements. Since the ...
4
votes
4answers
231 views

Find an integer $n$ such that $\mathbb{Z}[\frac{1}{20},\frac{1}{32}]=\mathbb{Z}[\frac{1}{n}]$.

How can we find an integer $n$ such that $\mathbb{Z}[\frac{1}{20},\frac{1}{32}]=\mathbb{Z}[\frac{1}{n}]$? Thanks in advance!
4
votes
1answer
290 views

Rings' second isomorphism theorem

I am thinking about the proof of the second isomorphism theorem, and something isn't very clear to me. Let $R$ be a ring ,$S\subset R$ a subring and $I\subset R$ an ideal. We have the natural ...
6
votes
3answers
465 views

Can you have a ring homomorphism from a ring to itself which isn't the identity?

By a ring I mean a ring with a multiplicative identity. To me, at this point, this sounds like a fairly simple question, but I haven't been able to come up with any such homomorphism, nor has ...
4
votes
1answer
945 views

Confusion between principal ideal and ideal

Artin defines an ideal $I$ as : $I$ is a subgroup of $R^+$ If $a \in I$ and $r \in R$ , then $ra \in I$ And Principal Ideal is defined as "In any ring, the set of multiples of a particular ...
1
vote
1answer
110 views

Matrix Rings over Artinian commutative Rings

Let $R$ be a commutative artinian ring with identity. It is true that for $n>0$ the matrix ring $M_n(R)$ is left and right artinian?
3
votes
3answers
280 views

Algebraic Elements and Fields of Quotients

The algebraic elements of $\mathbb{R}$ are those elements which are roots of nonzero polynomials with coefficients in $\mathbb{Q}$. In fact, by multiplying through by denominators, we can even take ...
3
votes
1answer
260 views

Characterize finite dimensional algebras without nilpotent elements

Characterize all finite dimensional algebras (may not be commutative) over a field $K$ without nilpotent elements. My condition: Let $A$ be any algebra (may not be finite dimensional), then it's ...
1
vote
1answer
111 views

Commutants of commutative algebras

Let $W$ be a unital algebra and let $V$ be its maximal abelian subalgebra. Must the commutant $V^\prime$ of $V$ be commutative?
3
votes
1answer
318 views

Intersection of principal ideals

An intersection of principal left ideals need not be principal but incidentally this phenomenon is witnessed in von Neumann regular rings. How about arbitrary intersections of infinitely many ...
2
votes
2answers
115 views

homomorphisms and product rings

The problem is this: Let $f:\mathbb{R}[x]\rightarrow \mathbb{C}\times \mathbb{C}$ be the homomorphism defined by $f(x)=(1,i)$ and $f(r)=(r,r)$, for $r\in \mathbb{R}.$ Determine the kernel and the ...
12
votes
4answers
744 views

Are these two quotient rings of $\Bbb Z[x]$ isomorphic?

Are the rings $\mathbb{Z}[x]/(x^2+7)$ and $\mathbb{Z}[x]/(2x^2+7)$ isomorphic? Attempted Solution: My guess is that they are not isomorphic. I am having trouble demonstrating this. Any hints, as to ...
1
vote
2answers
88 views

Small step in proving Hilbert's (Weak) Nullstellensatz

I'm reading up on a proof of Hilbert's Nullstellensatz which uses the Artin-Tate lemma. I followed all of it except for one step, which is probably quite elementary, but my brain may be too fried from ...
9
votes
2answers
470 views

A non-nilpotent formal power series with nilpotent coefficients

Does anyone have an example of a formal power series $$p=a_0+a_1x+ a_2x^2 + \cdots \in R[[x]]$$ ($R$ is a commutative ring) all of whose coefficients $a_i$ are nilpotent in $R$ such that $p$ is not ...
6
votes
2answers
182 views

Irreducible polynomial over field of order p

Let $p$ be a prime and $F=\mathbb{Z}/p\mathbb{Z}$ and $f(t)\in F[t]$ be an irreducible polynomial of degree $d$. I need to show that $f(t)$ divides $t^{(p^{n})}-t$ if and only if $d$ divides $n$.
3
votes
3answers
221 views

How to find ideals?

Let $R=\mathcal{C}([0,1],\mathbb{R})$ be the ring (standard one) of continuous functions. For each $\gamma\in[0,1]$, let $I_\gamma=\{f\in R; f(\gamma)=0\}$. It is easy to prove that $I_\gamma$ is an ...
4
votes
2answers
205 views

How to determine if this is a field?

A finite subring $R$ of a field $V$ contains $1$ (so $1$ is an element of $R$). The question is: True or False: The ring $R$ must be a field. I thought that if $R$ was a field it had to be a finite ...
2
votes
2answers
246 views

Noncommutative rings, finding $a$ and $b$ such that every term in the sum $(a+b)^n = a^n + a^{n-1}b + \ldots$ is distinct

This question was inspired by the binomial theorem for rings. For commutative rings, we have the identity $$(a+b)^n = \sum_{k=0}^n {n \choose k}a^kb^{n-k}$$ which does not hold for non-commutative ...
2
votes
1answer
64 views

For which $m \in \mathbb N$ is the ideal $(m,x^2+y^2)$ prime in $\mathbb Z[x,y]$?

Let $m \in \mathbb N$. Find a necessary and sufficient condition for $m$ such that the ideal $(m,x^2+y^2)$ is prime in $\mathbb Z[x,y]$. I have to find for which $m$ the quotient ring is an ...
4
votes
2answers
147 views

Localization and Extension of modules

Let $R$ be a commutative ring and $S$ be an $R$-algebra. Assume that $S$ is finitely generated as an $R$-module. Let $M$ and $N$ be finitely generated $S$-modules and $\mathfrak{m}$ a maximal ideal ...
8
votes
1answer
288 views

Tensor product of simple modules

Let $M$ a right simple module and $N$ be a left simple module over a ring $R$. My questions are: How can we describe $M \otimes_R N$ explicitly? Well, I guess that it is a quotient of $R$ by a sum of ...
1
vote
1answer
68 views

Whether there is a non commutative k-algebra of dimension no larger than 3

Suppose $k$ is a field and $A$ is a $k$-algebra of dimension no larger than 3. If $A$ is semi-simple, then $A$ can be written as a direct sum of simple $k$-algebras. Further one can find $A$ is ...
2
votes
1answer
59 views

How many products to be a ring?

I got the question below studying this problem: $p$-Sylow subring. Let $(R_1,+_1,\cdot_1)$ and $(R_2,+_2,\cdot_2)$ be two rings with identity elements $e_1,e_2$. Let $(R,+)$ be the group defined by ...
2
votes
1answer
217 views

Name a ring of 2 by 2 matrices where $a^3 = a$ and a belonging to this ring?

I need an example of a ring consisting of 2 by 2 matrices where $a^3=a$ with $a$ belonging to this ring. If someone can list the elements I would be satisfied. What I'm trying to get at it is ...
1
vote
2answers
225 views

How can we compute the power of an ideal?

Let $I$ be an ideal in a Noetherian ring $R$ which is generated by $x_1,...,x_n$. From this system, can we find out what is the generating set for an arbitrary power of $I$: $I^k$? Is it ...
4
votes
1answer
72 views

$p$-Sylow subring

I would like to know if there is some results concerning about the following question: When could a $p$-Sylow subgroup of a finite ring $R$ be a subring? In other words, is it possible to induce ...
-2
votes
1answer
118 views

Annihilator of a simple module 2 [duplicate]

Possible Duplicate: Annihilator of a simple module Let me ask the same question as before because I still have trouble understanding the problem. Let $R$ be a finitely generated ...