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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Polynomials' multiplicative inverse

Let $A(x)$ be a polynomial with integer coefficients. Is there always a polynomial $B(x)$ for which $$A(x)\cdot B(x)\equiv 1\pmod n$$ (for a given integer $n$). If the answer isn't yes, an answer ...
5
votes
1answer
245 views

Field Extension Notation

I've seen similar questions asked here, but I've not been able to find a comprehensive answer. I know that for a ring $R$, $R[X]$ denotes the ring of polynomials over $R$ and $R(X)$ denotes the field ...
1
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1answer
558 views

Is there a nice description of the field of fractions of the ring of polynomials with integer coefficients?

Let $\mathbb{Z}[x]$ denote the ring of polynomials (in the formal variable $x$) with integer coefficients. Since $\mathbb{Z}[x]$ is an integral domain, we can form its field of quotients, call it $Q.$ ...
3
votes
1answer
317 views

Describe $R=\mathbb{Z}[X]/(X^2-3,2X+4)$

I need to describe a ring $R=\mathbb{Z}[X]/(X^2-3,2X+4)$ I know that its element would be of the form $\{f(x)+(X^2-3,2X+4)|f(x)\in \mathbb{Z}[X]\}$ After that will i divide $f(x)$ by $X^2-3$ ...
1
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2answers
109 views

Chain conditions satisfied by some algebraic structures including ideals in certain commutative rings

From http://en.wikipedia.org/wiki/Ascending_chain_condition: "In mathematics, the ascending chain condition (ACC) and descending chain condition (DCC) are finiteness properties satisfied by some ...
10
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1answer
288 views

Generalizing the natural numbers - has this been studied before?

The ordinals numbers generalize the natural numbers and satisfy a generalized induction principle. However, the algebraic properties of the ordinal numbers aren't so good. For example, ordinal sums ...
3
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0answers
59 views

Field of fractions of the ring of trigonometric functions [duplicate]

I'm thinking about this question and no elegant idea comes to my mind. Prove that the field of fractions of $\mathbb R[x,y]/(x^2+y^2-1)$ is isomorphic to $\mathbb R (t)$.
4
votes
1answer
108 views

factorising polynomials

I got stuck on my algebra homework. First of all, can you check if the following statements are correct? The polynomial $f=3X^8+6X^4+2$ is irreducible in $\mathbb{Z}[X]$ because $f \in ...
5
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0answers
132 views

“Evaluation Homomorphisms” for Formal Power Series

In the ring of formal power series $\Bbb R[[x]]$ it is easy to check by induction that $$ 1 = (1-x)(1 + x + x^2 + \cdots). $$ Does this derivation imply the same identity for those real or complex ...
2
votes
1answer
138 views

Idempotent elements in any ring R.

I want to check this: if $e_1$, $e_2$ are idempotent elements in a ring $R$, then $Re_1=Re_2$ if and only if $e_1=e_2$.
2
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1answer
123 views

Generalised composition factors

Let $A$ be a semiprimary ring. A simple module $L$ is said to be a generalised composition factor of $M$ if there are $M'$ and $M''$, $M'' \subset M'$, submodules of $M$, such that $M'/M'' \cong L$. ...
1
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1answer
65 views

Why is $\phi: \ R[X] \rightarrow R[X]: \ \sum_{i=0}^n a_i X^i \mapsto \sum_{i=0}^n a_{n-i}X^i$ multiplicative?

I did the following exercise: Define $\phi: \ R[X] \rightarrow R[X]: \ \sum_{i=0}^n a_i X^i \mapsto \sum_{i=0}^n a_{n-i}X^i$. Let $f = \sum_{i=0}^n a_i X^i$ with $a_0 \neq 0 \neq a_n$. Show the ...
5
votes
1answer
201 views

number of element in a principal ideal domain can be $25/36/35/15$?

Could any one tell me number of element in a principal ideal domain can be $25/36/35/15$ ? I just know a principal ideal domain is generated by a single element. what the knowledge I need to find ...
2
votes
1answer
202 views

Quotient Rings of Polynomials Over Rings that are NOT fields

Let R be a commutative ring with identity and $f(x) \in R[x]$ be monic. In general, I am having trouble with identifying what $R[x] \over (f(x))$ is when R is not a field. What "tools" exist that can ...
3
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0answers
57 views

Weakening the assumption that an ideal is maximal

This question is from ChI, $\S{3}$ of Serge Lang's Algebraic Number Theory. Let $A$ be a commutative integral domain, integrally closed in its quotient field $K$, and let $E$ be a finite extension of ...
3
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2answers
123 views

Quotient field of a certain quotient ring

Let $A$ be a commutative integral domain and $\mathfrak p$ a prime ideal of $A$. Let $A_{\mathfrak p}$ be the localization of $A$ at $\mathfrak p$ and $\mathfrak{m}_{\mathfrak{p}}=\mathfrak ...
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0answers
41 views

Commutativity of s-unital rings.

Theorem. Let $R$ be a left (resp. right) s-unital ring. If $R$ satisfy $(P_1)$ (resp. $(P_2)$). Then R is commutative(and conversely). $(P_1)$ $y^{s}[x,\, y]=\pm x^{p}[x^{m},y^{n}]^{r}y^{q}$ where ...
4
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2answers
575 views

Cardinality of the quotient ring $\mathbb{Z}[x]/(x^2-3,2x+4)$

This problem is from a practice exam I was working on. What is the cardinality of the quotient $\mathbb{Z}[x]/(x^2-3,2x+4)$ ? Thoughts. If I find a ring that is easier to handle then this then I ...
2
votes
0answers
55 views

Product of ideals closed set

Let $R$ be a topological ring (in fact $R$ is metrizable) and let $I,J$ denote ideals of $R$. Suppose also that they are closed with respect the topology of $R$. Is it always true that the product ...
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0answers
138 views

Construction of reducible polynomials

We are given a polynomial $F(x)$. We are given enough coprime monic polynomials $p_{i}(x)$ that we are free to choose. Let $F_{i}(x) \equiv F(x) \pmod {p_{i}(x)} \neq 0$. Considering $F_i(x)$ as a ...
1
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1answer
76 views

An example of s-unital rings

I am just studying s-unital rings. A ring is called left (resp. right) s-unital if $x\in Rx$ (resp. $x\in xR$) for all $x$ in $R$. A ring is called s-unital if and only if $x\in xR\cap Rx$ for all ...
3
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1answer
96 views

Find a polynomial ring $R$ which is not an integral domain and an ideal $I$ such that $R/I$ is a field.

Question: Find a polynomial ring $R$ which is not an integral domain and an ideal $I$ such that $R/I$ is a field. Answer: $R=\mathbb{Z}_6[x]$, $I=\langle 2,x\rangle$, $R/I$ is isomorphic to ...
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4answers
407 views

$\mathbb{Q}[X,Y]/(Y^2-X^3)$ is not a UFD

I'm trying to show that $R = \mathbb{Q}[X,Y]/(Y^2-X^3)$ is not a UFD, but I got stuck. To prove this, I could try to find two "different" factorisations for one element, but I am not familiar with ...
1
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2answers
1k views

Give an example to show that a factor ring of an integral domain may be a field [duplicate]

A) Give an example to show that a factor ring of an integral domain may be a field B) Give an example to show that a factor ring of an integral domain have divisors of 0. C) Give an example to show ...
2
votes
1answer
105 views

Three quotient-ring isomorphism questions

I need some help with the following isomorphisms. Let $R$ be a commutative ring with ideals $I,J$ such that $I \cap J = \{ 0\}$. Then $I+J \cong I \times J$ $(I+J)/J \cong I$ $(R/I)/\bar{J} ...
4
votes
1answer
144 views

Finitely generated ring with zero Krull dimension

I'm trying to prove the following: Every finitely generated ring with Krull dimension equal to zero is finite. I'm trying to show that the ring is a domain, hence a field, in order to use the ...
2
votes
1answer
58 views

Invertibility of an element of the form $1+r+r^2+ \cdots+ r^{2005}$ in a ring when $r^n=0$ for some $n$

This is from a practice exam I was working on. In particular this isn't homework. Let $R$ be a ring with $1\in R$. Suppose that $r\in R$ satisfies $r^n=0$ for some positive integer $n$. Prove that ...
2
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2answers
56 views

$x^{n(x,y)} - x$ commutes with $y$, then $R$ is commutative

Theorem. If in a ring $R$ for every pair of elements $x$ and $y$ we can find an integer $n(x, y) > 1$ which depends on $x$ and $y$ so that $x^{n(x,y)}-x$ commutes with $y$, then $R$ is commutative. ...
5
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1answer
144 views

The uniqueness of a special maximal ideal factorization

The following problem is from Michael Artin's Algebra, chapter 12, M.6, unstarred: Let $R$ be a domain, and let $I$ be an ideal that is a product of distinct maximal ideals in two ways, say ...
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1answer
131 views

Principal Ideal Domain Basics.

Let $R$ be a Principal Ideal Domain and $a,b,c,d$ elements in $R$, such that $ab-cd=1$. I am trying to figure out why $Rb \cap Rd=Rdab+Rbcd$. In case this is true, I am wondering weather it is enough ...
3
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1answer
61 views

Is there a right semihereditary domain which isn't right Ore?

I do not have a lot of examples in my head for semihereditary domains at all, and I haven't been able to see how to resolve this question: Is there a right semihereditary domain which isn't right ...
0
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2answers
63 views

The set of all polynomials in $F[x,y]$ having constant term $0$ is an ideal, but not a principal ideal

In Fraleigh, it said, "Consider the set N of all polynomials in x and y in F[x,y] having constant term 0. Then N is an ideal, but not a principal ideal." (p.399) Could you tell me why this is ...
5
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1answer
222 views

Classification of all semisimple rings of a certain order [closed]

I'd appreciate it if you tell me where to begin in order to solve this question: Classify (up to ring isomorphism) all semisimple rings of order 720. Could the Wedderburn-Artin Structural Theorem ...
-1
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1answer
90 views

Idempotent elements in $(\mathbb{Z}_n,+,\cdot)$ [duplicate]

Can we find the idempotents in $(\mathbb{Z}_n,+,\cdot)$ for any $n$? Is there a general rule? Note: Trying to consider the prime factors!
4
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1answer
53 views

If $R$ is a rng, show that $R\times \mathbb{Z}$ contains a subset in one to one correspondence with $R$.

Let $(R,+,\cdot)$ be a rng (satisfies all the axioms of a ring except multiplicative identity). Define addition and multiplication in $R\times\mathbb{Z}$ by: $(a,n)+(b,m)=(a+b,n+m)$ and ...
2
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1answer
68 views

Some homework about ideals

I will ask you some things about ideals. Determine weither the ideal $(X^2+3) \subset \mathbb{F}_5[X]$ is maximal or prime. Intuitively I'd say that the ideal is prime but not maximal. To prove ...
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0answers
166 views

Equivalent conditions for commutativity in rings with 1

I have a problem with understanding the proof of theorem $3.1$. Could anyone help me with this? $(P_1)$ $y^{s}[x,\, y]=\pm x^{p}[x^{m},y^{n}]^{r}y^{q}$ where $m>1,r>0,n\geq0, s\geq0, p\geq0 , ...
4
votes
3answers
204 views

Non-unital module over a ring with identity?

Is there a nontrivial example of a non-unital module over a ring with identity? By trivial, I mean modules with $rm = 0$ for all $r$ and $m$. Just an idle question. (Why is there no tag for that?) ...
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0answers
49 views

An extremal combinatorial problem over Finite rings

Let $q$ be an odd number and $g_i = (g_{i1} g_{i2} \dots g_{ir}) \in \Bbb Z_q^r$ a list of vectors with $i\in\{1,\ldots,L\}$. Let each $g_i$ have $0 < k < r$ zero entries. What is the maximum ...
3
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3answers
429 views

Let $R$ be a Artinian commutative ring with $1 \neq 0$. If $I$ is prime, then $I$ is maximal.

Prove: Let $R$ a Artinian commutative ring with $1 \neq 0$. If $I$ is prime, then $I$ is maximal. I got stuck on this. I understand that every ideal is generated by finitely many elements. Here is ...
2
votes
2answers
100 views

ring isomorphism question

Could any one tell me how to solve this two? $1.$ As a ring $\mathbb{Z}[i]/(3-i)\cong\ ?$ $2.$ $L=\mathbb{R}[x]/(x^2-x+1),\ M=\mathbb{R}[x]/(x^2+x+1),\ N=\mathbb{R}[x]/(x^2+2x+1)$. Who is ...
2
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1answer
184 views

Are there any homomorphisms from integers into finite rings other than modulo $n$?

Are there any "homomorphisms" from $Z$ onto finite rings other than $Z/nZ$ ? I think if instead of mapping $k$ to $k$ (mod $p$), you map it to $p - (k$ (mod $p$)$)$ and you get $f(-ab) = f(a)f(b)$. ...
7
votes
3answers
408 views

The structure of a Noetherian ring in which every element is an idempotent.

Let $A$ be a ring which may not have a unity. Suppose every element $a$ of $A$ is an idempotent. i.e. $a^2 = a$. It is easily proved that $A$ is commutative. Suppose every ideal of $A$ is finitely ...
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0answers
51 views

Can you put a ring structure on the group of units modulo $n$? What *other* rings are there?

Let $n=4$ Then with multiplication defined as: $$ \begin{bmatrix} 0 & 1 & 3\\ 1 & 3 & 0\\ 3 & 0 & 1 \end{bmatrix} $$ i.e. $a_{i,j} = i \oplus j$. Then combined with the ...
2
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3answers
38 views

Localization and ultrapowers

I'm finishing up my senior thesis on ultraproducts and am trying to replace some of my longer proofs with nicer ones. Let $P$ be a prime ideal of $R$, and $F$ an ultrafilter on a set $X$. If $R_F$ ...
1
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2answers
244 views

Example of nil and commutator ideal

I have a problem with understanding a few definition. I can't find it anywhere. Does the set of all nilpotent elements is called nil? And is it always ideal? Could you give me some example of nil? Is ...
2
votes
2answers
86 views

Nonunits in a Noetherian Domain have an Irreducible Factor

I think I've proven the following statement without using the fact that it is a domain: Prove every nonunit in a Noetherian domain has an irreducible factor. Proof: Suppose we have a ring which ...
0
votes
1answer
92 views

discrete valuation ring

I am struggling to understand the proof of the following proposition Let $A=\{x\in K|v(x)\ge 0\}$ for a field $K$ be a discrete valuation ring. Let $t\in A$ s.t. $v(t)=1$. Then any element $x\in A$ ...
0
votes
1answer
41 views

Problem with understanding the proof of theorem. Let $R$ be a ring with unity satisfy condition $(P_1)$ . Then $N(R)\subseteq Z(R)$.

I have a problem with understanding this proof below. Could anyone elaborate on this, please? Theorem. Let $R$ be a ring with unity satisfy condition $(P_1)$: $P(x,\, y)=\pm Q(x,\, y)$, where $P(x,\, ...
3
votes
2answers
144 views

How to show that $\sqrt{\{0\}} = \bigcap_{\text{I is prime}}I$ in a commutative ring with unit

Here's another question about ring theory. How to show that $\sqrt{\{0\}} = \bigcap_{\text{I is prime}}I$ in a commutative ring with a unit. Own attempts For "$\subseteq$", let $x$ be an arbitrary ...