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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1answer
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Do real quadratic fields with unique primary factorization exist?

Bumped in Stillwell's book "Elements of Number Theory" into "The real quadratic fields with unique prime factorization are still not known ...". But doesn't $\mathbb{Q}[\sqrt{2}]$'s ring of integers ...
1
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1answer
26 views

Prove socle is ideal

In any ring $R$ define the socle as the sum of all minimal right ideals of $R$. Say we have two minimal ideals $A,B$. If $a\in A,b\in B$, then $a+b$ is in the socle. If $x\in R$, then ...
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3answers
31 views

Centre of matrix ring over skew field

Let $R$ be a semisimple ring. Show that $R$ is simple iff the centre of $R$ is a field. Book's solution: If $R$ is simple, it has the form $\mathfrak{M}_n(K)$ for a skew field $K$, and its ...
2
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1answer
25 views

What characterizes the equivalence classes of the quotient ring, P(N)/Fin(N)?

Let P(N) be the powerset of the natural numbers. Let Fin(N) be the collection of all finite subsets of N. Then (P(N),symmetric difference, intersection) is a ring. I am taking my first course in ring ...
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0answers
57 views

Existential theory

I am reading the following about (positive) existential theory: Could you explain to me the last sentence of the Lemma $1.6$ ? Why does this hold?
2
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3answers
147 views

Find what $A/S$ is isomorphic to, where $A = \Bbb Z_2[X_1,\dots,X_n]$ and $S=\langle X_1^2-X_1,\dots,X_n^2-X_n\rangle$

Find what $A/S$ is isomorphic to, where $A = \Bbb Z_2[X_1,\dots,X_n]$ and $S=\langle X_1^2-X_1,\dots,X_n^2-X_n\rangle$. I'm sorry but I don't have anything to add here. I've been trying it with ...
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1answer
29 views

How do I prove that the standard definition of prime ideal is equivalent to that of Krull's? [duplicate]

Definition Let $R$ be a commutative ring and $I$ be a proper ideal of $R$. Then $I$ is prime if and only if $\forall a,b\in I, a\in I$ or $b\in I$. Let $R$ be a commutative ring and $P$ be a ...
2
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0answers
53 views

A field F can be extended to one in which all polynomials over F have a solution

Please check to see if my methods are correct. This question is a lot to absorb all at once. Let P be the set of non-constant polynomials over a field F and let X be the set of finite subsets of ...
0
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0answers
34 views

Inverse limits of quotient rings

Let $A\subset B$ be an extension of discrete valuation rings and let $p$ and $P$ be the non-zero prime ideals of $A$ and $B$ respectively. So I can write $pB=P^m$ for some $m>0$. I form the ...
1
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1answer
13 views

Factorization Process in a polynomial ring

Reading the book "Field Theory" by S. Roman, in chapter $0$ I found the following problem: Let $F$ be a field and consider the polynomial ring $F[x_1,x_2,\ldots]$ where $x_i^2 = x_{i-1}$. Show that ...
1
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2answers
28 views

Radical of a ring [duplicate]

Let $A$ be a commutative ring with unity. Let the radical $\operatorname{Rad}(A)$ of $A$ be the ideal consisting of all nilpotent elements of $A$. Is $\operatorname{Rad}(A)$ of $A$ the same as ...
1
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1answer
17 views

Centre of matrix ring isomorphic to centre of ring

Show that the centre of $\mathfrak{M}_n(R)$ is isomorphic to the centre of $R$. Book's solution: If $A=(a_{ij})$ is in the centre of $R$, then $Ae_{rs}=e_{rs}A$, hence ...
0
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1answer
25 views

Looking for an example of an ideal contained in the union of other ideals, but not in any ideal individually

I'm looking for an example of the following scenario: $A, B, C $ are three ideals such that $C\subseteq A\cup B $ but $C\not\subseteq A $ and $C\not\subseteq B$. Any help would be great!
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2answers
31 views

Let $w$ be a primitve third root of unity. Find the units of $A=\{a+bw, a,b \in \mathbb{Z}\}$

What I have so far: if $x \in \mathbb{C}$, then $N(x)=\bar{x}x$ is multiplicative ($N(xy) = N(x)N(y)$). So $N$ restricted to $A$ is also multiplicative. if $a+bw \in A$, then it's easy to see that ...
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3answers
82 views

Should a ring be closed under multiplication?

In the definition of a ring, it is nowhere stated that it must be closed under multiplication. But it seems to be true for all the examples of rings that I've seen so far. So, is this implicitly ...
2
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2answers
54 views

Polynomial algebra and polynomial ring

What is the difference between polynomial algebra and polynomial ring? because sometimes I read polynomial algebra and it looks like a polynomial ring $K[x,y,..]$ in many variables. Thanks
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1answer
30 views

If $R$ is a simple Artinian ring, then when is a finitely generated module free?

Here's an exercise from my book, which only gives a brief solution which leaves me very confused. Let $R$ be a simple Artinian ring, say $R=K_r$. Show that there is only one simple right ...
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2answers
54 views

What is the significance of $A+ (B\cap C)=(A + B)\cap C$, where $A\subseteq C$, for modules?

My book (Introduction to Ring Theory, Paul Cohn) states this as a theorem and gives a proof. The book usually skips over trivial/easy proofs, so I don't really understand why this is in here. Isn't ...
2
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1answer
37 views

Show that $V(y^5-x^2)\subset \mathbb{R}^2$ is not isomorphic to $\mathbb{R}$ as a variety.

This is an exercise from Ideals, Varieties and Algorithms by Cox et al. Show that $V(y^5-x^2)\subset \mathbb{R}^2$ is not isomorphic to $\mathbb{R}$ as a variety by showing that there is no ring ...
2
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1answer
53 views

If a set $S$ generates an ideal $I\subset F[x_1,x_2,\ldots,x_n]$, then there is a finite subset $S_0 \subseteq S$ which generates $I$

The question: If $I$ is an ideal in $F[x_1,x_2,\ldots,x_n]$ generated by a set of polynomials $S$, then there is a finite subset $S_0 \subseteq S$ which generates $I$. By the Hilbert Basis ...
3
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0answers
56 views

Is ${(k^n)}^{\otimes r}$ a faithful $k\Sigma_r$-module for $n\geq r$?

I have the following question: Let $k$ be an infinite field. Let $E:={(k^n)}^{\otimes r}$ and let the symmetric group $\Sigma_r$ act from the right on $E$ by place permutations. It is well-known that ...
1
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1answer
53 views

Show that $R = \bigcap_mR_m$ whenever $R$ is an integral domain

Show that $R = \bigcap_mR_m$ whenever $R$ is an integral domain, where the intersection is indexed by all maximal ideals of $R$. $R \subset \bigcap_mR_m$ is clear since $R \subset R_m$ for all $m$ ...
5
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1answer
74 views

Question concerning a property of polynomial functions on $\Gamma:=\text{GL}_n(K)$ and the Schur algebra

I'm reading Green's book ''Polynomial Representations of GL_n. with an Appendix on Schensted Correspondence and Littelmann Paths''. Consider theorem 2.4b) part (i) on page 14: Consider the map $e : ...
3
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0answers
36 views

How to prove that the Schur algebra is isomorphic to a certain endomorphism ring?

I'm reading Green's book ''Polynomial Representations of GL_n. with an Appendix on Schensted Correspondence and Littelmann Paths''. Consider theorem 2.6c) on page 18: Let $K$ be an infinite field ...
2
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0answers
58 views

Question concerning a self-injective algebra and a faithful module

I'd like to know how corollary 2.11 of http://www.sciencedirect.com/science/article/pii/S002186930098726X# follows from theorem 2.10 from the same reference. 1) I know that $A$ being self-injective ...
0
votes
1answer
28 views

Given $A$-modules $N \subset M$ such that $N_m=M_m$ for all maximal ideals $m$, show that $M=N$

I am working on this exam question 6 $A$ is commutative ring with $1$ a) If $N \subset M$ are $A$-modules and $N_m=M_m$ for all maximal ideals $m$, show that $M=N$. We know that $N_m=M_m$ ...
3
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0answers
29 views

Short proof of the coincidence of left dom.dim., right dom.dim. and relative dom.dim. for semi-primary left QF-3 rings?

is there a short and/or elementary proof of the following fact (which is taken from theorem 7.7 from H. Tachikawa, ‘‘Quasi-Frobenius Rings and Generalizations’’, Springer Lecture Notes in ...
4
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2answers
61 views

Flat Non Projective $A$-Module [duplicate]

A standard fact in Commutative Algebra is that a Projective $A$-module is flat. The converse is false. Can someone show me an example of a Flat Non Projective $A$-Module? Thank you!
3
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1answer
38 views

Redundancy in the definition of Dedekind domain?

Is there a domain which is noetherian and whose nonzero prime ideals are maximal, but which is not integrally closed? This may be a silly question to experts. I ask because I think I have found ...
3
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0answers
52 views

Dominant dimension $\geq 2$ implies a certain double centralizer property

let $A$ be an Artin algebra and $M$ in $\mathfrak{mod}\ A$. Let $A$ be left-QF-3 with minimal faithful left ideal $Ae$. Then the following are equivalent: $\bullet$ $Ae$-dom.dim.$(A)\geq 2$ ...
2
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1answer
27 views

Question concerning a faithful module over an Artinian ring

Let $A$ be an Artinian ring and $M$ in $\operatorname{\mathfrak{mod}} A$. Is it true that $M$ is faithful if and only if there is an exact sequence of the form $0\rightarrow A \rightarrow M^r$ for ...
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2answers
42 views

Let $K=GF(2)$ and $p(x)= x^3 + x+1.$ Show that $p$ is irreducible in $K[x]$

Let $K=GF(2)$ and $p(x)= x^3 + x+1$ Show that $p$ is irreducible in $K[x]$ First of all am I right in interpreting: $$GF(2) = \mathbb Z / 2 \mathbb Z= \{ 0,1\}$$ So basically, $p(x)$ is a ...
3
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2answers
42 views

For an exact sequence $0\to M_1\overset{f_1}\to\cdots\overset{f_r}\to M_r\to0$ is it true that $l(M_i)-l(M_{i+1})=l(\ker(f_i))-l(\ker(f_{i+1}))$?

For an exact sequence $0\to M_1\overset{f_1}\rightarrow\cdots\overset{f_r}\rightarrow M_r\to0$ is it true that $l(M_i)-l(M_{i+1})=l(\ker(f_i))-l(\ker(f_{i+1}))$? $M_i$s are modules and $l(M_i)$ ...
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0answers
31 views

Why is there a $q_i$ such that $q_j|q_i$?

Let $q_i$, a sequence of of irreducible polynomials where $q_i$'s highest-order term has coefficient $c_n = 1$ (by the way, what's the right term to describe this property?) Anyhow, let's look at: ...
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1answer
58 views

Proof of direct sum of ideal class group of Neukirch book

In books Neukirch, Algebraic Number Theory. I don't understand. 1) Why there exists $a$ such that $a\equiv c \ \mod \mathfrak p $ and $a\in ca_{\mathfrak p}^{-1}a_{\mathfrak q}$ for $\mathfrak ...
3
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1answer
45 views

Local ring and isomorphism problem

I have a local ring $R$ with maximal ideal $\mathfrak{m}$. Fixing some $x\in\mathfrak{m}$, I want to show that $\mathfrak{m}^{k-1} \subset (\mathfrak{m}^k : x)$ and conclude that $R/(\mathfrak{m}^k : ...
2
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2answers
63 views

Example for an ideal which is not flat (and explicit witness for this fact)

I'm looking for an ideal $\mathfrak{a}$ of an commutative (possibly nice) ring $A$ together with an injective $A$-module homomorphism $M\hookrightarrow N$ such that the induced map ...
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2answers
47 views

Is this another way of stating the Chinese Remainder Theorem?

Assume that $I + J = R$. Let $a,b \in R$. Find an element $u$ of $R$ satisfying $a + I = u + I$ and $b + J = u + J.$ I want to work on this, but I feel there's some issue of a missing theorem I ...
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0answers
58 views

short exact sequence of algebras over a field

Let $A,B,C$ be algebras over a field $F$ ($F=\mathbb{Q}$ or $\mathbb{Z}/p$, $p$ prime). The height of $A$ is defined to be $$ \mathrm{height}(A)=\sup_{a\in A}\inf\{n(a)\in \mathbb{N}\mid a^{n(a)+1}=0 ...
1
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1answer
71 views

Any deeper “duality” between non-zero-divisors and units of a ring?

I'm reading Aluffi's algebra book at the moment -- specifically, I'm on the introductory rings/modules chapter. I noticed two interesting pieces of information: in a (not necessarily commutative) ...
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1answer
50 views

$k\left[x,\,y\right]/\left(y^{2}-x^{3}\right)$ and $k\left[x,\,y\right]/\left(y-x^{3}\right)$ are not isomorphic. [closed]

For field $k$, I want to show that $k\left[x,\,y\right]/\left(y^{2}-x^{3}\right)$ and $k\left[x,\,y\right]/\left(y-x^{3}\right)$ are not isomorphic. I met wall from the beginning. Is there are some ...
6
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2answers
146 views

Cancellation problem: $R\not\cong S$ but $R[t]\cong S[t]$ (Danielewski surfaces)

I would like to understand why the two rings $$ R={\mathbb{C}[x,y,z]}/{(xy - (1 - z^2))} \\ S=\mathbb{C}[x,y,z]/{(x^2y - (1 - z^2))} $$ are not isomorphic, but $R[t]\cong S[t]$. This example is ...
2
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1answer
18 views

Noetherian PI rings are fully bounded

Can any one give me a precise reference for justifying that Noetherian PI rings are fully bounded. Thanks a lot.
2
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3answers
77 views

What does the quotient group $(A+B)/B$ actually mean?

I understand that $A+B$ is the set containing all elements of the form $a+b$, wit $a\in A, b\in B$. When you do the quotient group, that's like forming equivalence classes modulo $B$. All elements ...
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5answers
43 views

An example of a ring homomorphism $f:R\to R'$ that is not onto, $I$ is an ideal of $R$ and $f(I)$ is an ideal of $f(R)$.

My ring theory professor proved that even if a ring homomorphism $f:R \to R'$ is not onto then the image of an ideal in $R$ is an ideal of $f(R)$. I understood the proof but I cannot think of an ...
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2answers
41 views

Are all injective endomorphisms of a module automorphisms?

As far as I understand, an automorphism is an isomorphism from a set to itself. If we have a homomorphism $f:M\rightarrow M$, then, from the first isomorphism theoreom, $im(f)$ is a submodule of $M$. ...
0
votes
1answer
40 views

What is the number of ways to express $\mathbb{Z_n}$, the ring of integers modulo $n$, as a direct sum of its ideals?

$\mathbb{Z}_n$ is a ring, ($\{0,1,2,...,n-1\}, \mod n$ addition and multiplication). I think that the ideals of $\mathbb{Z}_n$ are precisely the rings generated by its divisors. For example, the set ...
4
votes
1answer
71 views

Principal ideal domains that are not integral domains

In the usual definition, a principal ideal domain $R$ is also assumed to be an integral domain? However, the property that every ideal is generated by a single element does not seem to immediately ...
3
votes
1answer
64 views

Banach algebra with left or right minimal ideal without minimal bi-ideal

Let $A$ be a Banach algebra( or a ring). A left ideal $I$ of $A$ is called a left minimal ideal, if $I\neq\{0\}$ and there is no any other non-zero left ideal of $A$ completely lies in $I$. With a ...
1
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1answer
41 views

What is a prime ideal?

I am having some trouble understanding the concept of a prime ideal in ring theory. I have researched what a prime ideal is and the simplest answer I got was this: An ideal $P$ of a commutative ...