This tag is for questions about rings, which are a type of algebraic structure studied in abstract algebra and algebraic number theory.

learn more… | top users | synonyms (1)

1
vote
1answer
20 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 $IJ \subset P$ $IJ \subset I$ ...
1
vote
1answer
34 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 ...
2
votes
2answers
59 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
votes
0answers
25 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 ...
11
votes
1answer
700 views

On limits, schemes and Spec functor

I have several related questions: Do there exist colimits in the category of schemes? If not, do there exist just direct limits? Do there exist limits? If not, do there exist just inverse limits? ...
1
vote
1answer
82 views

Prime ideals in $R[x]$, $R$ a PID

Let $R$ be a PID. Show that if $\mathfrak p \subset R[x]$ is a prime ideal, $(r) = \left\{h(0) \colon h(x) \in \mathfrak p \right\}$, and $$\mathfrak p = (r, f(x), g(x)),$$ where $f(x), g(x) \in ...
0
votes
0answers
73 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 ...
1
vote
1answer
45 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
votes
1answer
53 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, ...
-2
votes
0answers
28 views

Indecomposable commutative rings [on hold]

Let $R$ be a commutative ring. Can we say that $R=\bigoplus_{i\in I}R_i$ or $R=\prod_{i\in I}R_i$ where $R_i$ are commutative ring and $I$ is an infinite set?
5
votes
2answers
53 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 ...
0
votes
0answers
34 views

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: ...
7
votes
2answers
463 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 ...
1
vote
0answers
42 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} ...
0
votes
1answer
43 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 ...
3
votes
1answer
36 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!
3
votes
2answers
58 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, ...
0
votes
0answers
29 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?
-1
votes
2answers
33 views

Ring homomorphism and ideal that contains the kernel [on hold]

If $f:R\rightarrow S$ is a ring homomorphism and $I$ ia an ideal of $R$ such that $ker(f) \subseteq I$ then $f^{-1}(f(I))=I$ We know that $I\subseteq f^{-1}(f(I))$ but how can I use that $ker(f) ...
0
votes
1answer
42 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 ...
2
votes
2answers
107 views

integral ring extension, maximal ideals

Let $\varphi:A\rightarrow A'$ be an integral ring extension. 1) Show that for every maixmal ideal $m'\subset A'$ the ideal $\varphi^{-1}(m')\subset A$ is maximal 2) and that for every maximal ...
0
votes
1answer
13 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 $\xi^n,\eta^n$? ...
14
votes
1answer
85 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 ...
1
vote
1answer
22 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 ...
1
vote
0answers
26 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 ...
2
votes
0answers
23 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 $ ...
0
votes
1answer
18 views

Using Exchange Lemma in a decomposition

The "Exchange Lemma" in the decomposition of modules states that any decomposition of right $R$-modules $M_1⊕...⊕M_n=A⊕B$ with the endomorphism ring $End (A_R)$ local yields $M_i≈A⊕X$ for some $i$, ...
2
votes
1answer
33 views

Computing injective hulls over a lower triangular matrix ring

Let $R$ be the ring $\begin{pmatrix} {\mathbb Z }_{ p } & 0 \\ {\mathbb Z }_{ p } & {\mathbb Z }_{ p } \end{pmatrix}$ with $p$ prime. $R$ is an artinian ring and is also a $\mathbb ...
5
votes
2answers
59 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 ...
6
votes
1answer
90 views

How to show that $\mathbb{C}[x_1,x_2,x_3, x_4]/(x_1x_2 - x_3x_4, x_1x_3 - x_2x_4, x_1x_4 - x_2x_3)$ is an integral domain?

I am looking for a way to show that the ring $\mathbb{C}[x_1,x_2,x_3, x_4]/I$ where $$I = (x_1x_2 - x_3x_4, x_1x_3 - x_2x_4, x_1x_4 - x_2x_3)$$ is an integral domain. In other words I want to show ...
3
votes
2answers
111 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 ...
0
votes
1answer
303 views

When Leavitt path algebras are unital

I’ve been taking a seminar course on Leavitt path algebras, and our source material is rather laconic as far as explanations are concerned. I am attempting to justify (to myself) the following claim ...
0
votes
2answers
66 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 $ ...
3
votes
1answer
32 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 ...
-2
votes
0answers
44 views

Chain of three prime ideals of $\mathbb F_p[x,y]$ [closed]

Let $A_1\neq\left \{ 0 \right \} $, $A_2$, and $A_3$ be prime ideals of $\mathbb{F}_p[x,y]$ such that $$A_1\subset A_2\subseteq A_3\subset \mathbb F_p[x,y]$$ Then $A_2 = A_3$.
0
votes
1answer
42 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 ...
4
votes
2answers
66 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 ...
3
votes
1answer
432 views

Correspondence theorem for rings.

Could someone provide a reference that includes a full and honest proof of the Correspondence Theorem for rings? Let $A$ be a multiplicative ring with identity and $I$ an ideal of $A$. There is a ...
0
votes
1answer
33 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, ...
0
votes
1answer
128 views

Is the quotient ring $\mathbb Z[x]/(5, x^3+x+1)$ a field?

The problem is this: Show that $A=\mathbb Z[x]/(5, x^3+x+1)$ is a field. I tried to show that that ideal is a maximal ideal, but failed. Since A is finite set, so it suffices to show that (5, ...
4
votes
2answers
233 views

Idempotents in a Quotient Ring

Let $R=\mathbb{Z}_p[x]/(x^p-x)$. Show that $R$ has exactly $2^p$ elements satisfying $r^2=r$. I know that for $f,g\in\mathbb{Z}_p[x]$, we have $f-g\in(x^p-x)$ if and only if $f(a)=g(a)$ for all ...
2
votes
3answers
234 views

Can we say “commutative ring = field”?

We know the difference between ring ($R$) and field ($F$) is that $R$ does not guarantee multiplication is commutative. Now, if considering commutative $R$, which means ($R$, $*$) is a group, can ...
1
vote
1answer
61 views

Regarding constructing the $I$-adic completion of a ring

Let $R$ be a commutative ring and let $I \subset R$ be an ideal. For $n \geq m$, let $\varphi_{m,n}$ denote the canonical ring homomorphism $R / I^n \to R / I^m$. Let $J = \cap_n I^n$. Then $R / J$ ...
4
votes
1answer
37 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
votes
1answer
104 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
votes
1answer
14 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 ...
7
votes
1answer
93 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 ...
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 ...
1
vote
3answers
74 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 $\overline{r}$?
1
vote
0answers
22 views

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 ...