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

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Additive exact functors preserve homology of modules

If $C$ is a chain complex of modules over a ring $R$ and $F: Mod_R \to Mod_S$ an additive exact functor then: If $F$ is covariant then $H_n(FC)\cong FH_n(C)$. Can anyone give my an idea or an ...
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1answer
69 views

Trouble showing flatness

Let $K$ be a field and $\pi: K[x]/(x^2) \to K$ be the ring homomorphism given by the valuation at $0$. I'm stuck in showing that $\pi^*(K)$ (the pullback) is not a flat module (over $K[x]/(x^2)$).
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1answer
27 views

Polynomial greatest common divisor algorithem

I look for an algorithm for Polynomial greatest common divisor. I saw this at Wikipedia but I didn't understand where is the algorithm. If you have other source for this algorithm, or you can write ...
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1answer
73 views

A problem with tensor products

Let $K$ be a field, $R=K[x^2,x^3]$, $S=K[x]$, and consider $S$ as an $R$-module. Given $f: S \to R \oplus R$ so that $f:p \mapsto (x^3p,-x^2p)$, prove that $f\otimes 1: S \otimes_R S \to (R\oplus R) ...
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21 views

Module and Noetherian/Artinian Rings

I am trying to prove that: Every finitely generated $F$-module $M$ is both Noetherian and Artinian where $F$ is a field. For this I am looking at the submodules of $F$ and saying that they are in ...
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2answers
39 views

A non-projective module

Let a ring with identity $R$ be decomposed as $S_1⊕\cdots⊕S_n$ (as a right $R$-module), where $S_i=e_iR$ with $e_i$ nonzero idempotents of $R$ adding up to $1$. If $J$ is the Jacobson radical of ...
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39 views

Equalities in the categories of modules

It is well-known that over a quasi-Frobenius ring $R$ any f.g. right module $M$ is reflexive, in the sense that whenever we take $M^*=Hom_R(M,R_R)$ as a left $R$-module, the modules $M$ and $M^{**}$ ...
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50 views

Behavior of the annihilators of modules via monic and epic homomorphisms

Let $f:M\rightarrow N$ be an $R-$ homomorphism. Prove that if $f$ is monic, then $l_{R}\left(M\right)\supseteq l_{R}(N)$ , whereas if $f$ is epic, then $l_{R}(M)\subseteq l_{R}(N)$ . This is ...
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1answer
46 views

Finitely generated modules [duplicate]

Suppose that $R$ is a commutative ring and $M$ and $N$ are finitely generated $R-$modules. What we can say about $Hom_R (M,N)$? is it a finitely generated $R$-module?
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29 views

showing Module is simple

Given the following: let $C \subset \mathbb{H}$ be a subring of the real quarternion algebra such that it contains the center of $\mathbb{H}$ = $Z(\mathbb{H})$ Also C $\cong \mathbb{C}$ Then let R ...
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arithmetic modulo field with real numbers vector space

if i choose modulo3 to be the field, and the real numbers to be the vector space. how do i Multiplier vector in scalar? for example i take "4" from the vector space of real numbers and want to ...
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1answer
21 views

How to determine $Hom (M,M)$ for an irreducible $R$-module $M$?

More exactly, I'm considering $R$ to be a finite dimensional $\mathbb C$-algebra. For any $R$-module $M$, the $\mathbb C$ vector space $Hom_R(M,M)$ contains scalar multiplication and hence contains ...
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39 views

Equivalence of Modules

The category of modules over a ring can be viewed as an enriched version of an action of a monoid on a set (see nLab entry). Moreover, if $R$ is a commutative ring, the category of modules over it is ...
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1answer
26 views

Finding all $\mathbb{Z}$-submodules of $\mathbb{Z}$ with a complement

I am trying to find all the $\mathbb{Z}$-submodules of $\mathbb{Z}$ that have a complement Here with complement I mean: Def: Let $N \subset M$ be an $R$-Submodule. A submodule $N'$ is called ...
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3answers
61 views

What is the meaning of $x*\sqrt{3} \mod 1$?

What is the meaning of $x*\sqrt{3} \mod 1$? I'm trying to understand this: $$5( x*\sqrt{3} \mod 1) $$ If we talk about: $x=19,22,48,98$ what will be the result? I don't know how to calculate ...
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47 views

Finitely generated module over PID

Let $R$ be a PID and let $p$ be a prime ideal in $R$. (a) Let $M$ be a finitely generated torsion $R$-module. Use that $p^{k-1}M/p^kM \cong F^{n_k} $ where F is the field $R/(p)$ and $n_k$ is the ...
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1answer
12 views

homomorphism of product of modules

Given $(A_i)_{i\in I}$ and $B$ modules over a ring $R$, with $I$ infinite, $A_i\ne 0\;\forall i$ and $B\ne 0$, is it true that $$Hom_R(A_i,B)=0\;\forall i\implies Hom_R(\Pi_{i\in I} A_i, B)=0\quad?$$ ...
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1answer
13 views

prove that $HOM_R(\bigoplus _{s\in S}R_s ,M)$ and $\prod_{s\in S}M_s$ are isomorphic as $R$ modules.

Let $R$ be a ring, $M$ module over $R$ and $S$ a non empty set. Let $\mathrm{Hom}_R(M,N)$ be the group of $R$ module homomorphisms from $M$ to $N$. Denote $M_s:=M$ and $R_s:=R$. Prove that the ...
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29 views

What is the kernel of $I/I^2 \to \Omega_{\mathbb P^{n}/k} \otimes \mathcal O_X$?

Recall that if $X \subset \mathbb P^n$ is a smooth projective variety, we have the conormal sequence of locally free sheaves on $X$ (here $I$ is the ideal sheaf of $X$): $$ I/I^2 \xrightarrow{\delta} ...
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1answer
61 views

When are two simple tensors $m' \otimes n'$ and $m \otimes n$ equal? (tensor product over modules)

Suppose that $M$ is a right R-module and $N$ is a left $R$-module. We can construct $M \underset{R}\otimes N$ and give it an Abelian group structure by considering the free R-module $K$ generated by ...
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57 views

References about “algebras over monoids”

Please, could someone point me any reference (with a bit of details) about "algebras over monoids" (in the sense of Schwede & Shipley, Algebras and modules in monoidal model categories)? Thank you ...
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44 views

Are projective modules “graded projective”?

Let $A^{\bullet}$ be a graded commutative algebra. Denote by $A^{\bullet}$-mod the category of graded modules over $A^{\bullet}$. Let $A$ be $A^{\bullet}$ considered as an algebra (we forgot grading). ...
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1answer
35 views

Example of non-commutative ring with exactly 2014 two sided-proper ideals.

find a non-commutative ring with exactly 2014 two sided-proper ideals.find a ring with exactly 2014 pairwise non-isomorphic irreducible modules. If it was the commutative ring i would have thought of ...
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1answer
29 views

Difference between to Tensor products with regards to modules

What would be the difference between $$ \otimes_B $$ and $$ \otimes $$ both in the following context and in general? Let A be a ring with $$ B \subset A $$ and M a B-Module. We can construct the ...
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2answers
55 views

Show that $\operatorname{Hom}(S(-d),S)\cong S(d)$ where $S$ is polynomial ring?

As stated above, $S$ is polynomial ring, and since the polynomial ring is $S$ and $S(-d)$ are finite over $S$ as graded modules, we can say that $\operatorname{Hom}(S(-d),S)$ is also graded. My ...
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1answer
40 views

Direct sum of simple modules and Schur's Lemma

Suppose $M,N$ are two non-isomorphic simple $R$-modules. For $m,n\geq1$, is it true that $$ \text{Hom}_R(M^{\oplus m},N^{\oplus n})\cong\hat{0}\,? $$ I think it's true by Schur's Lemma. ...
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1answer
42 views

A finite-dimensional $K$-algebra?

Let $A$ be a commutative noetherian domain of characteristic $p>0$ and $K$ be its quotient field. Let $G$ be any finite group whose order is divisible by $p$. Is $KG$ a finite-dimensional ...
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51 views

Looking for help, Aluffi Exercise 5.13, Chapter 6: characterization of PIDs

I quote: "Let $M$ be a finitely generated module over an integral domain $R$. Prove that if $R$ is a PID, then $M$ is torsion-free if and only if it is free. Prove that this property characterizes ...
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43 views

Trouble on Aluffi's Exercise 5.5, Chapter 6: finitely generated projective module over a local ring is free

The overall goal of the problem is the following: Let $R$ be a local commutative ring (so that there is a unique maximal ideal $\mathfrak{m}$) and let $M$ be a finitely generated $R$-module. Assume ...
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1answer
38 views

What is ker $f \otimes g$?

I am trying to find out the $ker(f\otimes g)$ where $f:M \rightarrow P$ and $g:N \rightarrow Q$ are $A$ linear maps where $A$ is not a field.So $(f\otimes g):M\otimes N\rightarrow P\otimes Q $ is $A$ ...
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1answer
36 views

Isomorphism with tensor product

Let $f$ : $\Bbb Z_2$ $\rightarrow$ $\Bbb Z_4$ given as $f$($a$ + $2$$\Bbb Z$) = $2a$ + $\Bbb Z_4$ is a monomorphism . And knowing that <0,2> as subgroup of $\Bbb Z_4$ is isomorphic to $\Bbb Z_2$ ...
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1answer
51 views

Is a vector of coprime ring elements column of an invertible matrix?

Given a commutative ring $R$ with unit and $a_1=(r_1,\ldots,r_n)^T \in R^n$ with coprime entries (i.e. $\sum_i Rr_i=R)$. Are there $a_2,\ldots,a_n \in R^n$ such that the matrix $A = ...
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18 views

Rational Canonical Form Confusion; Choosing Basis Which Gives the Rational Canonical Form.

I am reading the theory of finitely generated modules over a PID. One of the applications of the the theory is that one can derive the theory of rational canonical form of a linear operator on a ...
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64 views

Why $R^{op}$ used in the definition of “category of $R$-modules”?

Earlier today, I was thinking: "Oh, an $R$-module is just an additive functor $R \rightarrow \mathbf{Ab}.$" Anyway, I had a bit of a read over at nLab, and it says: For any small ...
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22 views

Center of a ring projective?

If $R$ is a ring and $Z(R)$ denotes $R$'s centre, then when is $R$ projective as an $Z(R)$-module?
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1answer
16 views

Proof that the connecting morphism in the snake lemma is well defined

The snake lemma says: suppose we have two exact sequences of $R$-modules $M_1 \xrightarrow{f_M} M_2 \xrightarrow{g_M} M_3 \rightarrow 0$ $0\rightarrow N_1 \xrightarrow{f_N} N_2 \xrightarrow{g_N} ...
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27 views

If $M$ is a $R$-module with $R=R_1\times \cdots \times R_n$ then $M\simeq M_1\times \cdots \times M_n$ where $M_i$ is a $R_i$-module?

Let $R_1, \ldots, R_n$ be rings and consider the ring $R:=R_1\times \cdots\times R_n$. How can I show every $R$-module $M$ is isomorphic to a product $M_1\times \cdots\times M_n$ where each $M_i$ is a ...
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1answer
33 views

A group ring has finite length

I have been given a version of Maschke's Theorem to prove: Let $k$ be a field and $G$ a finite group s.t. $|G|$ is non-zero in $k$. Show that the group ring $k[G]$ is a semi-simple ring. A hint ...
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48 views

Socle of submodule relative to the module

in these notes i am reading i am told that the socle of $K$ (where $K \subset M$ , and $M$ is a module) is = $K \cap$ Soc $ M$ But why is this? i see the intuition but cannot formalize a proof ...
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54 views

Homomorphisms of semi-simple submodules

I managed to show $\forall$ R-Module M, $\exists$ unique semi-simple submodule $sM \subset M$ containing every semi-simple submodule of M, by showing that the direct sum of semi-simple submodules are ...
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31 views

$I^{n-1}/I^n$ is a Noetherian $R$-module

Let $R$ be a commutative ring with unity and $I \subseteq R$ an ideal. Prove: if $R/I$ is a Noetherian ring and $I/I^2$ is a finitely-generated $R$-module, then $I^{n-1}/I^n$ is a Noetherian ...
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14 views

All the solutions for this system 5x+33y = 6 (mod 13) and 7x + 2y = 9 (mod 13)

I want all the solutions for this system. 5x + 3y = 6 (mod 13) and 7x + 2y = 9 (mod 13)... Thanks
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1answer
19 views

$f:M\to N$ is surjective $R$-linear map

$R$ be a commutative ring with $1$ and maximal ideal $m$.Let $f:M \rightarrow N$ be a $R-linear $ map and $N$ is finitely generated $R-$ module.Suppose that the induced homomorphism $M\otimes_R(R/m) ...
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38 views

When considering the commutative ring $R$ over itself, if its submodule is free, then the submodule equals to the module?

When considering the commutative ring $R$ over itself, then this $R$-module is isomorphic to $R$, but if $I$ is an ideal of $R$, then it is a submodule, if this submodule is free too, then it is ...
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21 views

When is the Tensor product of Modules itself a Module?

If $M$ is a right $R$ module and $N$ is a left $R$ module then $M \otimes_R N$ is an abelian group. Wikipedia says that if $M$ is an $R$ bimodule then $M \otimes_R N$ can take on the structure of a ...
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Show that field of fraction of a commutative domain is an indecomposable module which is not finitely generated

I came across this problem and get stuck for quite sometime. Problem: Let $R$ be a commutative domain that is not a field. Let $F$ be its field of fractions. Show that $F$ is an indecomposable ...
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38 views

Showing the socle of a module exists and is unique

I have been set the following exercise: For every $R$-module $M$, show that there exists a unique semi-simple submodule $sM$ $\subset$ $M$ which contains every semi-simple submodule of $M$. ...
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15 views

Ring of Upper triangular matrices submodule

Suppose that $R$ is the ring of all matrices of the form $ \left( \begin{array}{cc} z & p \\ 0 & q \end{array} \right) $, where $z\in \mathbb{Z}$ and $p,q\in \mathbb{Q}$. Let $$ I_n= ...
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2answers
65 views

Direct summand of modules proof

I'm have problems trying to show this, any help would be appreciated. Show $i: N \subset M$ is a direct summand iff $\exists$ a module map $r: M \to N$ s.t $ri = 1_{N}$, and that any complement of ...
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68 views

Is $R$ PID if every submodule of a free $R$-module is free?

Let $R$ be a commutative ring. Before I proved that every submodule of a free $R$-module is free over a P.I.D. Now I'm trying to prove the reciprocal, if every submodule of a free $R$-module is ...