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
38 views

Why do we need injectivity in the definition of integral dependence?

Let $f: A \rightarrow B$ a ring morphism of commutative rings, then one has on $B$ a multiplication by elements of $A$ defined by $b*a \doteq b.f(a)$ (where . is the multiplication in the ring $B$). ...
0
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1answer
71 views
2
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2answers
53 views

Ring Theory: Identity Elements

A brief question$:$ If a ring is specified to have an identity, is it implicit that the identity in question is the multiplicative identity? From the definition of a ring $R$, $R$ must contain ...
1
vote
1answer
20 views

Show that in ascending Loewy series, $S^r(R)=R$

Let $R$ be an Artinian ring, $N$ its radical, and $r$ the smallest natural number such that $N^r=0$. Define an ideal $S^n(R)$ of $R$ recursively as follows: $S^1(R)=soc(R)$ Assuming ...
7
votes
1answer
77 views

Proving Things About Rings Using Things About Vector Spaces

All rings below are assumed to be commutative and having an identity. $\newcommand{\bw}{\bigwedge}\newcommand{\R}{\mathbf R}\newcommand{\mc}{\mathcal}$ Consider the following problem: Problem 1. ...
1
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1answer
42 views

Understanding this proof regarding maximum ideals

Prove: Let $R$ be a commutative associative ring with $1$ and $M \triangleleft R$. The factor-ring $R/M$ is a field iff $M$ is the max ideal in $R$. ($\implies$) Let $R/M$ be a field, then ...
4
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1answer
74 views

Generalization of the derivative to polynomial rings

It is easy to see why the derivative plays an important role in real and complex analysis from the geometric viewpoint. However, one can extend the definition of a derivative to polynomial rings such ...
0
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1answer
31 views

Is the ideal $\{2m + (1 + \sqrt{-6})n:m, n\in\mathbb{Z}\}$ principal in $\mathbb{Z}[\sqrt{-6}]$?

Is the ideal $\{2m + (1 + \sqrt{-6})n:m, n\in\mathbb{Z}\}$ principal in $\mathbb{Z}[\sqrt{-6}]$? I have an exercise that asks just that. As a hint it says to prove that this ideal contains $1$, ...
4
votes
0answers
71 views

What properties $R \subseteq S$ should have in order that every prime ideal of $S$ is extended?

My question is almost the same as In what conditions every ideal is an extension ideal?; I allow myself to ask this question, since there is no answer to the above question. My question: Given ...
2
votes
2answers
40 views

Equivalent definitions of faithful flatness

I am currently studying the equivalent definitions of faithful flatness over (probably noncommutative) unital rings. In particular, there is a version that I have doubts about: Let $R \subseteq S$ ...
3
votes
1answer
67 views

Intuition behind $k$-algebra, $k$-algebra morphisms?

I will state the definition of a $k$-algebra and $k$-algebra morphisms. A ring $A$ equipped with a ring homomorphism $k \to Z(A)$ is called a $k$-algebra. More explicitly, this means that $A$ has ...
2
votes
1answer
35 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
78 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 ...
3
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1answer
41 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
93 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 ...
1
vote
2answers
61 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
24 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
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0answers
39 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
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1answer
35 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
votes
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
vote
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 ...
1
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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
votes
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
58 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 ...
-1
votes
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
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
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
vote
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
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
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
votes
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!
3
votes
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 ...
0
votes
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
votes
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
1
vote
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 ...
1
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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
votes
1answer
38 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
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
votes
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
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
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
votes
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
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
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
votes
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 ...