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
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1answer
33 views

A question about 1.0.3 in Grothendieck's EGA

In (1.0.3), Grothendieck states that, given non-commutative rings $A$ and $B$, a homomorphism $\varphi : A \to B$, and a left ideal $\mathfrak{J}$ of $A$, the left ideal $B\mathfrak{J}$ of $B$ ...
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6answers
76 views

Constructing $\mathbb{C}$ from $\mathbb{R}$

I'm having difficulty grasping the notion that you can define the complex numbers as $\mathbb{C}=\mathbb{R}[t]/\langle t^2+1\rangle$. As far as I understand, $\mathbb{R}[t]$ is the set of all ...
2
votes
1answer
37 views

If $R$ is a simple ring, is every corner $eRe$ simple?

Assume that $R$ is a ring, not necessarily commutative nor unital. Let $e$ be an idempotent in $R$. Is there a correspondence between ideals of $R$ and ideals of the corner ring $eRe = \{ere : r\in ...
3
votes
1answer
76 views

If $A$ is a finitely generated $R$-module, is $\operatorname{Hom}_R(A,R)$ finitely generated? [duplicate]

Let $R$ be an utterly arbitrary commutative, unital ring. Let $A$ be a finitely generated $R$-module. Is $\operatorname{Hom}_R(A,R)$ finitely generated as an $R$-module? Intuitively and based on ...
0
votes
1answer
57 views

Module of constant rank over noetherian reduced ring

Let $A$ be a reduced noetherian commutative ring and $M$ be a finitely-generated $A$-module such that for all prime ideals $\mathfrak p$, $M_{\mathfrak p}/\mathfrak pM_{\mathfrak p}$ is an ...
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2answers
60 views

Why do nil ideals annihilate simple modules?

A nil ideal $N$ of a ring $R$ is defined as follows: $(N,+)$ is a subgroup of $(R,+)$ $\forall x \in N, \forall r \in R :\quad x \cdot r \in N$ $\forall x \in N, \forall r \in R : \quad r \cdot x ...
0
votes
0answers
22 views

Looking for a Coordinate Free Way to Prove a Precursor to Nakayama Lemma.

Let $M$ be a finitely generated module over a ring (commutative with identity) $R$. Let $\mathfrak a$ be an ideal of $R$ and $\phi:M\to M$ be an $R$-module homomorphism such that ...
4
votes
0answers
37 views

Too Many Members in a Finitely Generated Module are Linearly Dependent

I am new to module theory and as of now am not very comfortable with the subject. So can somebody please check whether my claim and its proof is okay? Consider the following statement: Let $M$ be ...
0
votes
1answer
34 views

If $F$ is a field, then any two algebraic closures are isomorphic by an isomorphism that is the identity on $F$.

To start, suppose $K_1$ and $K_2$ are two algebraic closures of $F$. (a) Let $P$ be the set of partial functions $f$ from $K_1$ to $K_2$ with the following properties: $F$ is contained in ...
0
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1answer
26 views

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
25 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
29 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 ...
-1
votes
1answer
43 views

Find the number of units in $M_2(\mathbb Z_p)$ where $p$ is prime. [duplicate]

Find the number of units in $M_2(\mathbb Z_p)$ where $p$ is prime. I want a detailed solution, not just the number. $M_2$ means matrix of order $2\times 2$. I know the defn of units. But how to ...
2
votes
1answer
24 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
53 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
votes
3answers
146 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 ...
-1
votes
1answer
28 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
votes
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
votes
0answers
33 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
12 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
vote
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
vote
1answer
16 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
votes
1answer
24 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!
3
votes
2answers
28 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 ...
0
votes
3answers
69 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
votes
2answers
53 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
1
vote
1answer
29 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 ...
1
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2answers
51 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
votes
1answer
36 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
votes
1answer
52 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
votes
0answers
55 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
vote
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
votes
1answer
72 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 : ...
2
votes
0answers
35 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
votes
0answers
53 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
votes
0answers
28 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 ...
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votes
2answers
59 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
votes
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
votes
0answers
46 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
votes
1answer
25 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 ...
1
vote
2answers
40 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
votes
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 $l(M_i)$ ...
1
vote
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: ...
1
vote
1answer
57 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
votes
1answer
44 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
votes
2answers
59 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 ...
1
vote
2answers
46 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 ...
1
vote
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
vote
1answer
70 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) ...