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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when $ht_SQ =ht_Rf^{-1}(Q)$?

Let $R$ and $S$ be commutative rings with identity, and $f:R\to S$ be a homomorphism. By what assumptions we can have $ht_SQ =ht_Rf^{-1}(Q)$, for every prime ideal $Q$ of $S$? What about $ht_SQ ...
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1answer
34 views

Quotients in a non-discrete valuation ring

Let $R$ be a valuation ring, $\mathfrak{m}$ the maximal ideal of $R$. Let $k$ be the residue field, and $K$ the field of fractions of $R$. Assume that the valuation on $K$ is such that ...
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1answer
34 views

How to show that a certain module is injective over an endomorphism algebra?

Let $A$ be a self-injective Artin algebra and $M\in\ \mathfrak{mod}\ A$ with the property $\mathfrak{add}\ _AA = \mathfrak{add}\ M$. Let $I$ be a finitely generated injective $A$-module. Why is ...
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1answer
65 views

Isomorphism theorem for modules

Let $p$ be a prime and $n$ be a positive integer such that $p^n > 2$. Let $R:= \mathbb{Z}/p^n \mathbb{Z}$ and suppose $(x,y) \in R \times R$, where addition and multiplication are defined ...
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1answer
28 views

Commutative rings of characteristic 2

Is there a "canonical" form to write the commutative rings of characteristic $2$? For instance, I hoped that every such ring is isomorphic to ${\cal P}(X)$ with symmetric difference as addition and ...
2
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1answer
16 views

Does every ID with subring which has a unity have an unity?

For an arbitrary ID (integral domain) $R$ with subring $S$, assume that $S$ has an unity. Then does $R$ have a unity too? If not, please provide a counter-example.
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2answers
36 views

Adjoining elements to a ring

I am working in Adkins & Weintraub, and they ask the reader to consider the integers adjoined with a fraction $\dfrac{p}{q}$. I'm not really sure what they mean by this, since they didn't mention ...
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45 views

Finite colength ideals in a power series ring

$\newcommand{\Hilb}{\operatorname{Hilb}}$Let $I\subset R:=\mathbb{C}[[x,y]]$ be an ideal. Then, $I$ is said to be of colength $n$ if $\dim_\mathbb{C}(R/I) = n$. For example the ideal $(x,y)$ has ...
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0answers
27 views

Why do I get homogenizations of polynomials by trying to find roots in $\mathbb Q$.

I noticed that if I have a polynomial equation in, say $x$ that needs to be solved in $\mathbb Q$, one tactic is to substitute $x=y/z$ where $y$ and $z$ are coprime integers, but then after clearing ...
2
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2answers
39 views

For $I,J$ ideals $P$ Prime ideal, show that $IJ\subset P \iff I\cap J \subset P$

Question : Prove the following equivalence $IJ\subset P \iff I\cap J \subset P \iff$ $I$ or $J \subset P$ I was able to do this $IJ \subset I$ and $IJ \subset J$ so $IJ \subset P$ $IJ \subset I$ ...
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1answer
60 views

Maximal Ideals of $\mathbb C[x, y]$

I recently learnt that the maximal ideals of $\mathbb C[x, y]$ are precisely the ones of the form $(x-a, y-b)$ for some $a, b\in \mathbb C$. I am unable to prove it. So I considered an easier ...
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2answers
181 views

Commutative binary operations on $\Bbb C$ that distribute over both multiplication and addition

Does there exist a non-trivial commutative binary operation on $\Bbb C$ that distributes over both multiplication and addition? In other words, if our operation is denoted by $\odot$, then I want the ...
2
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0answers
33 views

Maximal ideals of R[x]/(f(x))

I have been studying for my Qualifying Exam and came across the following problem: Let $R\subseteq T$ be integral domains and suppose that $a\in T$ satisfies a monic polynomial of degree $d$ with ...
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1answer
48 views

Is every algebraic integer a sum of roots of $x^n - a$?

A complex number is said to be an algebraic integer if it is a root of a monic polynomial with integer coefficents. For example any root of the polynomial $x^n - a$ for $a \in \mathbb{Z}$ is an ...
2
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1answer
61 views

Riemann-Roch Theorem and Ideals of a Ring

I found in some Math book a comment stating that the study of Ideals in ring theory à la Dedekind (all kinds of ideals? only one-sided ideals?) could be transferred to other areas (specifically, ...
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2answers
467 views

Is ideal an “anti-field”?

I am comparing theorems on normal subgroup and ideal from Fraleigh's, and come to this strange intuition. I hope my conclusion does not screw up, I hope I won't get ridiculed: Theorem 15.18: $M$is ...
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0answers
50 views

Minimal Free Resolutions

Matsumura, Commutative Ring Theory, Chapter 7 p. 153-4: Let $(A, \mathfrak{m}, k)$ be a local ring. An exact sequence $$(*) \cdots \rightarrow L_i \xrightarrow{d_i} L_{i-1} ...
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0answers
86 views

Zero-Sum Partitions of Nonzero Elements of a Ring

In this question, rings are not necessarily finite nor do they need to be unital (i.e., the multiplicative identity may not exist). Although I shall almost exclusively discuss finite commutative ...
0
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1answer
44 views

Some doubts about right ideals of a ring

I would like to know whether the following paragraph regarding right ideals and modules is correct. Any comment or help is welcome: A right ideal of $R$ is just a submodule of the right $R$-module ...
5
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2answers
59 views

Ideals of non semi-simple group rings.

I worked for a long time on complex group rings and complex twisted group rings. In those cases the algebra is semi-simple and its structure is well understood from the decomposition to irreducible ...
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Converse of Chinese Remainder Theorem

Chinese Remainder Theorem for commutative rings with identity Let $R$ be a commutative ring with identity. If $I, J$ are ideals of $R$ satisfying $I+J=R$, then there is an isomorphism of rings: ...
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1answer
38 views

Decomposition of a homogeneous polynomial

Let $k$ be a field. Suppose I have a homogeneous polynomial $f$ in $k[x,y,z]$. If $f$ is reducible, does it always decompose as a product of homogeneous polynomials? Thanks!
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0answers
32 views

Under what conditions is a ZG-module torsion-free?

If we have a ZG-module A, I was wondering if there are known condition we may imply on either A or the group G to make A torsion-free?
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1answer
48 views

Proof about the difference between right and left ideals in a ring

I have tried get a version of the proof stating that a left ideals of a ring is not, in general, a right ideal, and viceversa. Is my formulation right? Comments and corrections are welcome. I have ...
0
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1answer
17 views

Perfect-power Gaussian integer factorization

In $\mathbb Z[i]$, consider a relation $\alpha\beta=\epsilon\gamma^n$ for $\epsilon$ a unit and $(\alpha,\beta)=1$. Then why are each of $\alpha,\beta$ associated to nth powers $\xi^n,\eta^n$? ...
3
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2answers
59 views

Kernel of ring homomorphism

Let $\phi: R \to R'$ be a ring isomorphism and $I$ an ideal of $R$. Define $\phi(I)=\{\phi(i): i \in I\}$. Show that $\frac RI \cong \frac {R'}{\phi(I)}$. To use the first isomorphism theorem, ...
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1answer
97 views

Is the ring of holomorphic functions on $S^1$ Noetherian?

Let $S^1={\{ z \in \Bbb{C} : |z|=1 \}}$ be the unit circle. Let $R= \mathcal{H}(S^1)$ be the ring of holomorphic functions on $S^1$, i.e. the ring of functions $f: S^1 \longrightarrow \Bbb{C}$ which ...
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1answer
24 views

Generalization of a Result on Modular Inverses

Yesterday, I attempted to solve the general system of linear congruences (I'm not sure why I've never tried this before.) \begin{align*} x &\equiv a \pmod{A} \\ x &\equiv b ...
2
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0answers
33 views

tensor products of Hopf algebras

Let $H_1,\cdots, H_n$ be Hopf algebras over the field $\mathbb{Z}_p$, $p$ prime. Then the tensor product $$ \bigotimes_{i=1}^nH_i $$ is still an algebra over $\mathbb{Z}_p$. Is $ \otimes_{i=1}^nH_i $ ...
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2answers
61 views

Finding the kernel of maps between (polynomial) rings

If I have a map between rings like $f\colon k[x_1,x_2]\to k[t],x_1\mapsto t^2-1,x_2\mapsto t^3-t$, how can I prove that the kernel is $\mathfrak{a}=(x_2^2-x_1^2(x_1+1))$? I see that ...
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1answer
142 views

Prove that $n \in \mathbb{Z}^\star \Leftrightarrow n \mid 1$ and $n-1 \mid 1$ or $n+1 \mid 1$, and $(x-1)/(t-1) \equiv n \pmod {t-1}$

I want to prove the following lemma: Let $F[t,t^{-1}]$ be the ring of the polynomials in $t$ and $t^{-1}$ with coefficients in the field $F$ and assume that the characteristic of $F$ is zero. ...
3
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1answer
33 views

localization of the Pontrjagin ring of an $H$-space with respect to $\pi_0$

Let $X$ be an $H$-space. Let $F$ be a field. Then $H_*(X;F)$ is a Hopf algebra over $F$. According to group-like elements in the Hopf algebra of the homology of H-spaces, $$ \pi_0(X)=\{g\in ...
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2answers
71 views

Prove that an ideal $ \mathfrak{m} $ of a commutative ring $ R $ is maximal iff $ R/\mathfrak{m} $ is simple.

Could someone give me a hint on whether I’m on the right track or not? For sufficiency, I tried the following: Suppose that $ \mathfrak{m} $ is a maximal ideal. With the quotient map, we get $ ...
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1answer
47 views

What's a diagonal sum of two matrices?

Let $A$ be an $n\times n$ matrix over a field $K$. Show that there exists an invertible matrix $P$ such that $P^{-1}AP$ is a diagonal sum of an invertible matrix and a nilpotent matrix. (Hint: use ...
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2answers
69 views

“If $x$ is a non-unit, then $1-ux$ is a unit”

I don't understand these two lines from my book. We are given that $R$ is a local ring. If every $2$-generator submodule is cyclic and $Ra$, $Rb$ are given, then $Ra+Rb=Rc$, hence ...
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1answer
35 views

What's a $2$-generator submodule?

Context: Let $R$ be a local ring and $M$ an $R$-module. Show that the set of all submodules of $M$ is totally ordered by inclusion iff every finitely generated submodule of $M$ is cyclic or, ...
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1answer
40 views

group-like elements in the Hopf algebra of the homology of $H$-spaces

Let $X$ be an $H$-space with product $\mu$. Then $H_*(X)$ is a Hopf algebra with product $\mu_*$. Let $\psi$ be the coproduct of the Hopf algebra $H_*(X)$. Define a subset $S$ of $H_*(X)$ as $$ ...
3
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1answer
121 views

Subring of $M_7(\mathbb{Z}_2)$ isomorphic to $\mathbb{F}_{128}$?

Let $A \subset M_7(\mathbb{Z}_2)$ be a subring such that no proper nonzero subgroup $V \subset \mathbb{Z}_2^7$ is invariant under all matrices in $A$. I suspect that $A \cong \mathbb{F}_{128}$, but ...
0
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1answer
15 views

solution verification: find characteristic of integral domain under given conditions

Okay, so this seems an easy problem, but I was having doubts if my solution was correct or not. I would really appreciate if somebody could verify it for me. Suppose $R$ is an integral domain such ...
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0answers
28 views

Conjecture concerning involutions in a unitary braided fusion category/Grothendieck ring

Despite the categorical setup, a solution to this question may require no categorical tools (see Conjecture 2). Let $\mathcal C$ be a unitary braided fusion category, $I$ be its set of isomorphism ...
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3answers
78 views

What does $\overline{r}m:=rm$ mean?

On this Wikipedia article, it says that you can define an $R$-module $M$ as an $R/Ann_R(M)$-module using the action $\overline{r}m:=rm.$ What does that action actually mean? What is $\overline{r}$?
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Decompose finitely generated modules and use Krull-Schmidt theorem [duplicate]

I'm trying to show that if $R$ is an Artinian ring, then for finitely generated modules $M,N,N'$, we have that $M\oplus N\cong M\oplus N'$ implies that $N\cong N'$. I'm supposed to do this by ...
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2answers
159 views

Modules that have finitely many submodules

Drawing the lattice of submodules of a given module helps me to gain some intuition about the structure of module. Sometimes, however, it is not possible to draw in neat manner; For example vector ...
3
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2answers
54 views

Constructing DVR's from arbitrary UFD's

Is the following statement true? Let $A$ be an UFD and $p\in A$ prime, then $A_{(p)}$ is a discrete valuation ring. I think yes: For every element $x$ of $Q(A_{(p)})=Q(A)$, there is a unique ...
3
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0answers
52 views

What properties do the rings of infinite, upper-triangular matrices have?

I'm very familiar with the ring of $n\times n$ matrices over a field, the ring of $n\times n$ upper triangular matrices over a field, and the ring of infinite column-finite matrices over a field. But ...
8
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1answer
101 views

different definitions of Hopf algebras

(i). In the book Algebraic Topology, A. Hatcher, p. 283, the notion Hopf algebra is defined as follows: (ii). However, in the book Bialgebras and Hopf algebras, J.P. May, the notion Hopf algebra is ...
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0answers
49 views

Nilpotent minimal ideals are one sided minimal ideals

Let $R$ be an Artinian ring. I am looking for some condition under which all nilpotent (two sided) minimal ideals of $R$ are both minimal left and minimal right ideals. Can anyone give some hint? ...
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1answer
102 views

When $N \to M \otimes_R N$ is not an embedding.

Can someone please provide an example of the following (or tell me why such an example doesn't exist): Let $R$ be a (not necessarily commutative) ring, $M$ an $R$-$R$-bimodule containing a copy of ...
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1answer
34 views

Ideals for non commutative ring

For a non commutative ring without identity, is it possible that there will be right and left ideals which are different?
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3answers
36 views

Finding Linear Combination of Polynomials

I am stuck on a question involving finding the greatest common divisor of polynomials and then solving to find the linear combination of them yielding the greatest common divisor. My work thus far is ...