For questions about modules over rings, concerning either their properties in general or regarding specific cases.

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Proof that $f:\mathbb CG \to S$ given by $f(a)=a\cdot s$ is surjective

I am going through a proof of the following result from my lecture notes: Suppose $\mathbb C G \cong \bigoplus_{i=1}^nU_i$ where the $U_i$ are simple $\mathbb CG $ modules. Then any simple $\mathbb ...
2
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0answers
13 views

Character of action of permutations on a subset

I have a problem involving characters of a certain $\mathbb C$$G$-module, where $G = S_n$. The module is the vector-space $V$ with basis $\{v(I) | I\subset\{1,...,n\},|I| = k\}$, where $k\le n$. With ...
0
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1answer
15 views

Castelnuovo-Mumford regularity of a shifted module

Let $R$ be a graded ring and $M$ be a graded $R$-module. Regularity definition $\operatorname{reg}(M)=\max\{j-i\mid \beta _{i,j}(M) \not=0\} $. What is the relation between ...
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1answer
13 views

every projective module has a free complement.

I have to prove that every projective module has a free complement. Now Rotman ask it to first do for $R=Z/6Z$ and $P=Z/2Z$, We know $Z/6Z \cong Z/2Z \oplus Z/3Z$. Now $Z/2Z$ is projective as it a ...
2
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1answer
23 views

Depth comparison on short exact sequences

Let $R$ be a Noetherian ring and $M,N,U$ be $R$-modules. We have a short exact sequence $$0 \longrightarrow U \longrightarrow M \longrightarrow N \longrightarrow 0.$$ We know that ...
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1answer
19 views

Why is $\text{End}_K(K) = K$, or, more generally, $\text{End}_D(D) = D^{\text{op}}$?

Let $K$ be a field and a vector space over itself, $D$ a division ring with a modul over itself, with its opposite ring $D^{\text{op}}$. Why is then $\text{End}_K(K) = K$? Or more general, why is ...
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1answer
10 views

Is the image of a ring homomorphism an ideal?

Let $\phi : R \rightarrow S$ be a ring homomorphism, then is $im(\phi)$ an ideal in $S$? I ask this because I am studying about modules and in that we say that for a given $R$-module homomorphism the ...
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1answer
13 views

Relation between splitness of short exact sequences and free modules

I just proved a result , that, a short exact sequence with a free module at the third position is split exact. i.e. The short exact sequence, $0\to \mathbf M' \to \mathbf M \to \mathbf M'' \to 0 $ ...
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1answer
12 views

Question on the argument proving primary decomposition theorem

Lang - Algebra p.150, Lemma 7.6 Let $E$ be a torsion module of exponent $p^r(r\geq 1)$ for some prime element $p$. Let $x_1\in E$ be an element of period $p^r$. Let $\bar E = E/(x_1)$. Let ...
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1answer
58 views

An exercise using Nakayama's lemma.

Let $A$ be a ring and $\mathfrak a \subseteq A$ an ideal. Let $N \to M$ be a homomorphism of $A$-modules such that the induced homomorphism $N/\mathfrak a N \to M/\mathfrak a M$ is surjective. If $M$ ...
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1answer
35 views

Prove that $I_k \otimes_k \Omega \rightarrow I$ is injective

Let $\Omega$ be an algebraically closed field, $k$ a subfield of $\Omega$, $I$ an ideal of $\Omega[X_1, ... , X_n]$, and $I_k = I \cap k[X_1, ... , X_n]$. Then $I_k$ is an ideal of $k[X_1, ... , ...
2
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1answer
16 views

Presented matrix and number ring.

Let $V$ be the module generated by the column matrix $A= (2, 1+ \sqrt{-5})^T$. Prove that the residue of $A$ in $\mathbb{Z}[\sqrt{-5}]/ \mathfrak{P}$ has rank 1 for every prime ideal $\mathfrak{P}$ of ...
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2answers
37 views

Existence of projection in proof of Maschke's theorem

In a proof of Maschke's theorem, my lecturer writes "If $N$ is a $\mathbb C G$-module and $N\leq M $ is a submodule, let $\pi: M \to M$ be a projection (i.e. a map with $\pi^2=\pi$) with image $N$." I ...
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2answers
48 views

An exercise on tensor product over an integral domain.

This post is the natural conclusion of another one (An exercise on tensor product over a local integral domain.). Let $M$ be a finite module over an integral domain $A$. Let $Q$ be its fraction ...
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0answers
45 views

Wolfram Mathematica [on hold]

Does somebody know how to write module in Wolfram Mathematica which will receive boolean function without knowing how variables it will have? Thanks.
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0answers
64 views

Let $R$ be a domain. Then $\operatorname{Tor}_n^R(A,B)$ is a torsion module

I have some problem to understanding the proof of this problem. This theorem is on page $414$ introduction to homological algebra Rotman. The theorem says: If $R$ is a domain, then ...
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1answer
111 views

Euler's theorem: [3]^2014^2014 mod 98

Calculate without a calculator: $$\left [ 3 \right ]^{2014^{2014}}\mod 98$$ I know I have to use Euler's Theorem. As a hint it says I might need to use the Chinese Remainder theorem too. I know ...
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2answers
45 views

How to show fraction field is flat (without localization)

Here I asked that if one can prove the field of fraction of a domain is flat. The answers used localization, which I am not familiar with. Can anyone prove it without using localization?
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2answers
38 views

Why field of fractions is flat?

I want to show this lemma: Let $R$ be a domain. If $A$ is a torsion $R$-module, then $\operatorname{Tor}_1^R (K,A)\cong A$ where $\operatorname{Frac}(R)=Q$ and $K=Q/R$. When I was reading ...
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1answer
51 views

Injection and surjection over free modules.

Let $A$ be a commutative ring and $M$ an $A$-module. Suppose to have both an injection $A^s \to M$ and a surjection $A^s \to M$ of module homomorphisms. Show that $M \simeq A^s$. This point is ...
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1answer
19 views

Why is a finitely generated Torsion-free module free?

Lang-Algebra p.148 Let $R$ be a PID. Let $M$ be a finitely generated Torsion-free $R$-module. Let $\{v_1,...,v_n\}$ be a maximal set of elements of $M$ among a given finite set of generators ...
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1answer
22 views

Matrix Ring of a Semisimple Ring

I recently read the concept of semi-simplicity of a (not necessarily commutative) ring. A ring $R$ is said to be semi-simple if $R$ as a left module over itself is a semi-simple module (This in turn ...
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2answers
37 views

Linearly independent elements are less than generators in a module.

Let $R$ be a commutative ring and $M$ a finitely generated $R$-module. Let $s$ the maximum number of linearly independent elements of $M$, while $t$ is the minimum number of a system of ...
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1answer
16 views

How to complete the diagram so that it becomes commutative?

Consider the following diagram of left $R$-modules ($R$ is a ring with identity $1_R$) and $R$-linear maps with exact lines: where $M$ is free. How do define $R$-lineas maps $M\longrightarrow ...
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1answer
25 views

Is every submodule of a free Noetherian module free?

Let $R$ be a ring and $M$ be a free $R$-module and $N$ be an $R$-submodule of $M$. If $M$ is Noetherian, is $N$ free? If $R$ is a principal ideal domain, then $M$ is Noetherian and $N$ is ...
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37 views

How much information about $R-\mathrm{Mod}$ can be extracted from $\underline{R-\mathrm{Mod}}$ and $K_0(R)$?

The question is in the title, so let me just repeat it: How much information about $R-\mathrm{mod}$ can be extracted from $\underline{R-\mathrm{mod}}$ and $K_0(R)$? Here ...
1
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1answer
59 views

An exercise on tensor product over a local integral domain.

Let $M$ be a finite module over a local integral domain $(A,m)$. Let $k$ be its residue field and $Q$ its fraction field. Consider the $k$-vector space $M \otimes_A k$ and the $Q$-vector space $M ...
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2answers
41 views

$\mathbb K[s, t]/\langle s-t\rangle\simeq \mathbb K[t]$ as $\mathbb K$-algebras?

How can I show there is an isomorphism of $\mathbb K$-algebras (where $\mathbb K$ is a field): $$\mathbb K[s, t]/\langle s-t\rangle\simeq \mathbb K[t]?$$ Above $\mathbb K[s, t]$ and $\mathbb K[t]$ ...
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37 views

Every module over a division ring is free?

I am currently trying to answer the following true/false question: True or False: Every module over a division ring $R$ is free. I know the result would be true if $R$ is a field (ie a ...
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20 views

Quotient of graded rings by graded ideals are again graded rings

I want to prove the following statement. Let $A = \bigoplus_{n=0}^{\infty}A_n$ to be a graded $R$-algebra and $I$ a graded ideal of $A$. Let $(A/I)_n = (A_n + I)/I$ be the image of $A_n$ in $A/I$. ...
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28 views

R has IBN but R fails rank condition [closed]

I need an example about "IBN for ring": R is a ring (no commutative), R has IBN but R fails rank condition. thanks
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Prove that the “additive” operation of the module($\mathbb{Z}_{p}^{*},\mathbb{Z}_{p-1},\cdot$) is continuous.

Consider the following module $\mathcal{M}=$($\mathbb{Z}_{p}^{*},\mathbb{Z}_{p-1},\cdot$) in which the "additive" operation is defined by normal multiplication in $\mathbb{Z}_{p}^{*}$ and scalar ...
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2answers
20 views

Simple generator modules

Let a ring $R$ with identity element be such that the category of left $R$-modules has a simple generator $T$. My question: "Is $T$ isomorphic with any simple left $R$-module $M$?" I tried the ...
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2answers
28 views

Why is a submodule of a free module over a PID is free?

Rotman - Advanced modern algebra p.650 Theorem 9.8 Let $R$ be a PID and $M$ be a free $R$-module and $N$ be an $R$-submodule of $M$. Let $\beta$ be an $R$-basis for $M$ and well-order it. Now, ...
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3answers
27 views

$||x-2|-3| >1$, then $x$ belongs to which interval

$||x-2|-3| >1$, then $x$ belongs to: (a) $(-\infty, -2) \cup (0,4) \cup(6,\infty)$ (b) $(-1,1)$ (c) $(-\infty, 1)\cup (1, > \infty)$ (d) $(-2,2)$ Answer:(a) My solution: let $|x-2| = p,$ ...
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1answer
21 views

The annihilator induces a module [duplicate]

Let $R$ be a ring, and $M$ an $R$-leftmodule. Let $\operatorname{Ann}_R(M)$ be the annihilator of M, meaning that $r m = 0 \space\space\space\space \forall r \in \operatorname{Ann}_R(M), m \in M$. ...
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1answer
20 views

Homomorphisms inbetween factor modules

Consider the Ring $\mathbb{Z}$ and the two ideals $(n), (m)$, where $n, m \in \mathbb{N}$, and consider $GCF(m, n)$ (the greatest common factor of $m, n$). Let p: $\mathbb{Z}/(m) \to \mathbb{Z}/GCF(m, ...
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2answers
30 views

If $I$ and $J$ are ideals of an $R$-algebra $A=I+J$ then $I\oplus J\simeq A\oplus (I\cap J)$?

Let $I$ and $J$ be two left ideals of an $R$-algebra $A$ such that $A=I+J$. Here $R$ is commutative ring with identity $1_R$. How can I show $$I\oplus J\simeq A\oplus (I\cap J)?$$ I've tried several ...
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Injective homomorphism on tensor product

I am currently attempting the following: Find (cyclic) $\mathbb{Z}$-modules $M, N, P$ and an injective homomorphism $f: M \rightarrow N$ s.t. $g: M \otimes_{\mathbb{Z}} P \rightarrow N ...
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Submodules $H$ satisfying: “if $ax \in H$ for some non-zero scalar $a$, then $x \in H$.”

Suppose $R$ is a commutative ring and that $X$ is an $R$-module. Question. Is there a term for those $R$-submodules $H$ of $X$ satisfying the following? For all $x \in X$, if $ax \in H$ ...
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21 views

Is there another terminology to designate this?

Let $R$ be a principal ideal domain. Let $M$ be a finitely generated $R$-module. Then there exists a free $R$-submodule $F$ of $M$ such that $M=Tor(M)\oplus F$ and the ranks of such $F$'s are the ...
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1answer
21 views

Simple modules over R isomorphic to R/I

Let $R$ be a ring, and let $M$ be a simple $R$-Module, meaning that it only has the trivial submodules {0} and $M$. Show that there's a maximal ideal $I \subset R$ so that $M \cong R/I$. Thanks in ...
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21 views

Zorn's Lemma Application for Finding Maximal Submodule?

I came across the following exercise in my algebra textbook. Let $M$ be a left $R$-module ($R$ is ring with unity $1_R$) and let $N\subseteq M$ be a submodule such that $x\in M-N$ for some $x\in ...
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27 views

Relation between two definitions of primary modules

Let $A$ be a commutative ring, $M$ be an $A$-module and $N \leq M$. There are two definitions of primary modules: 1) $M/N$ is coprimary (i.e., every zero divisor is nilpotent); 2) $\text{Ann}_A(N)$ ...
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1answer
26 views

Are there terminologies distinguishing these two ranks?

Definition 1. Let $R$ be a ring with invariant basis number and $M$ be a free $R$-module. Then, the rank of $M$ is the cardinality of an $R$-basis of $M$. ${}$ Definition 2. ...
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34 views

One-dimensional submodules of $\mathbb{C}^4$; direct sum of submodules.

I'm having some trouble understanding my lecture notes. I need to find a one-dimensional submodule $U$ of $\mathbb{C}^4$. Is $u=1+x+x^2+x^3$ valid? My reasoning: because $ux = x + x^2 + x^3 + 1 = u ...
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2answers
42 views

Is $m$ a projective $A$-module?

$A$ is a Noetherian local ring and $m$ be its maximal ideal. Then is $m$ a projective $A$-module? I got this problem while solving another problem. Can anyone please help me to figure it out?
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3answers
147 views

Is there a counterexample for the claim: if $A \oplus B\cong A\oplus C$ then $B\cong C$?

Here $A$, $B$ and $C$ are $R$-modules. Is there a counterexample for the claim: if $A \oplus B\cong A\oplus C$ then $B\cong C$? And what if $B$ and $C$ are finitely generated?
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0answers
40 views

some exterior power of a lattice, torsion-free? (about a remark of Serre in Local Fields)

I have a question on a remark of Serre in chapter III §2 of his book Local Fields (p. 49). In this section he wants to define the discriminant of a lattice with respect to a bilinear form. The ...
3
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1answer
44 views

Prove that the following is a non zero tensor.

I'm asked to prove that the ideal $I=(x,y)$ in $R=k[x,y]$ is not a flat R-module. My approach was to use the exact sequence $$0\rightarrow I \to R \to R/I \to 0$$ to induce a non injective map ...