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

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2
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1answer
60 views

Free modules are generators

Let $A$ be a ring. We say an $A$-module $M$ is a generator if for every $A$-module $N$ there is a surjective $A$-homomorphism $M^{(I)} \rightarrow N$ where $M^{(I)}$ denotes the $|I|$ copies of the ...
0
votes
2answers
191 views

What's a Dual Basis?

If $L/K$ is a finite extension of fields with $v_1, ... , v_n$ a basis, then we have an isomorphism of $K$-modules $L/K \rightarrow Hom_K(L,K)$, where a basis for $Hom_K(L,K)$ is the "dual basis" of ...
3
votes
1answer
111 views

Prove that $J$ is an injective module.

Suppose $J$ is an $R$-module, and for every cyclic $R$-module $C$ this short exact sequence splits $$0\rightarrow J\overset{f}{\rightarrow}B\overset{g}{\rightarrow}C\rightarrow 0.$$ Prove that $J$ ...
6
votes
1answer
257 views

Existence proof of the tensor product using the Adjoint functor theorem.

Can one prove the existence of the tensor product by the adjoint functor theorem? (of, say, modules over a commutative ring) If yes, how would one check the SSC (solution set condition) for the hom ...
3
votes
2answers
44 views

The computation of an inverse limit of modules.

I am trying to read the book Euler System by Karl Rubin. But I found something confusing, right at the page $1,$ or the page $9$ in the book. There we find that: Let $\mathcal O$ be the ring of ...
8
votes
2answers
228 views

Counterexample: multiplying modules by elements of an ideal vs. taking linear combinations

Let $R$ be a ring (commutative, unital) and $M$ an $R$-module. Let $I \subset R$ be an ideal. We make the following definitions: $$ A := \{ am \ | \ a \in I,\ m \in M \} $$ $$ B := \left\{ ...
1
vote
1answer
20 views

Terminology for a type of indecomposable module

Let $R$ be a ring. Is there a name for an $R$-module that is indecomposable, and each of its quotients is also indecomposable?
0
votes
1answer
29 views

$P$-generated module and exact sequence

For any $P$-generated module $N$, the canonnical exact sequence $P^{(\Lambda)}\to N\to0$ remains exact under $Hom_{R}(P,-)$ Why is this true? and is there any good reference to get more in this ...
0
votes
3answers
161 views

well defined mapping

Which of the following mappings is well-defined? a) $f: \mathbb{Z}_m \rightarrow \mathbb{Z}_m, \overline{x} \mapsto \overline{x^2}$ b) $g: {\mathbb{Z}}_m \rightarrow {\mathbb{Z}}_m, \overline{x} ...
1
vote
1answer
64 views

Proving a maximal element of a certain set of ideals is prime.

Let $R$ be a commutative ring and let $M$ be a nonzero $R$-module. If $m\in M$, define $\operatorname{ord}(m) = \{r \in R \mid rm = 0\}$, and define $F = \{\operatorname{ord}(m) \mid m \in M\ \wedge m ...
1
vote
1answer
113 views

Invariant Factor Decomposition of Integer Matrices

Suppose $M$ is an $n \times n$ matrix with integer entries. If we think of $M$ as acting on $V=\mathbb{Q}^n$, then we can think of $V$ as a $\mathbb{Q}[M]$-module and have that $V$ is isomorphic to a ...
3
votes
1answer
404 views

Torsion-free and projective modules over a Dedekind domain

Suppose that $A$ is a Dedekind domain (and integral domain). I am trying to prove that if $M$ is a torsion-free $A$-module, then it is projective and vice versa. Suppose that $M$ is projective. Then ...
2
votes
1answer
118 views

T. Y. Lam's definition for the rank of a projective module

This is a concept from Lectures on Modules and Rings of T. Y. Lam. It's on page 34-35 of the book. Throughout this post, $R$ will be used to denote a commutative ring. Let $P$ be a f.g ...
2
votes
1answer
89 views

A noetherian ring is self-injective if and only if all projectives are injective

Let $R$ be a left and right noetherian ring. I try to prove that the following two statements are equivalent. $R$ is injective as a left $R$-module All projective left $R$-modules are injective In ...
2
votes
1answer
120 views

Indecomposable module over a PID, Jacobson's Basic Algebra I - followup

I am trying to solve the question which already appeared here: Indecomposable $D$-module $M$, Jacobson's Basic Algebra I. Let $D$ be a PID. Let $M$ be a finitely generated $D$-module. Then $$ M\ ...
2
votes
1answer
84 views

Free modules and free submodules. [duplicate]

Just finished my semester in advanced abstract algebra and there was one question that I could not answer. Let $R$ be commutative and let $(e_1,...,e_n)$ be a base for $R^{(n)}$. Put ...
2
votes
2answers
110 views

If $R[s], R[t]$ are finitely generated as $R$-modules, the so is $R[s + t]$.

Let $S \supset R$ as rings with $1 \in R$. Suppose that $s, t \in S$ and that the subrings $R[s], R[t]$ are finitely generated by $\{1, s, \dots, s^k\}$ and $\{1, t, \dots, t^m \}$. Then $R[s + ...
2
votes
2answers
53 views

Certain submodules of a finitely generated $R$-module are finitely generated?

Let $R \subset S$ be commutative rings. Let $s \in T$ where $T$ is a finitely generated $R$-module in $S$ with $R \subset T \subset S$. I want to show that $R[s]$ is finitely generated as an ...
0
votes
0answers
32 views

System of linear equations with modulo elements

Transform the following $3 \times 3$ matrix $A$ with entries in $\mathbb{Z}/5\mathbb{Z}$ into reduced row echelon form (with $[1]$ as Pivot element and $[0]$ above the Pivot element). State out the ...
1
vote
1answer
52 views

Sufficient condition for simple module

Every non-zero module homomorphism $f:M\rightarrow N$ is injective. Prove that $M$ is simple. $f$ is not the zero map so $M$ is not $\{0\}$. I'm guessing I should let $N$ be a proper submodule of ...
0
votes
1answer
56 views

For every positive integer $i$, a free module $F$ with basis $B_1$ over a ring $R$ with identity has a basis $B_{i+1}$ with $|B_{i+1}|=|B_1|+i+1$.

Let $F$ be a free module over a ring $R$ with identity. Suppose $F$ has two finite basis $B_1,B_2$ such that $|B_2|=|B_1|+1$. Prove that for every positive integer $i$, $F$ has a basis $B_{i+1}$ ...
0
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3answers
103 views

Question about tensor product of modules, when does $c \otimes r = 0$?

Let $R$ be an integral domain, and let $K$ be its field of fractions. Let $C$ be an $R$-module. Then is $C \cong C \otimes_R R$ injectively contained in $C \otimes_R K$? Define $\phi: C \otimes_R ...
1
vote
0answers
62 views

Modules over dedekind domain

Let $L$ be a finitely generated module over a dedekind ring $R$ and let $M$ be a submodule of $L$ such that $L/M$ is finite. I want to describe $L/M$. For a prime ideal $\mathfrak{p}$ of $R$, let ...
2
votes
1answer
69 views

Injective module over a quotient ring

I want to make sure I am not totally confused about this. Suppose $R$ is an integral domain, $I$ a (non-zero) ideal of $R$. Then if $M$ is a (non-zero) $R/I$-module is it necessarily not injective ...
4
votes
1answer
93 views

Classifying torsion-free injective modules over a PID

So the question is as in the title: Let $R$ be a PID. Classify all torsion-free injective modules. I know that it is going to be divisible, and using torsion free, if we define ...
1
vote
0answers
55 views

Show that we have an algebra homomrohpsim

I need to show that we have an algebra homomorphism $\phi: M_n(K)\otimes_KA \simeq M_n(A)$ Where A is a K-algebra and K is some field. I suspect it's really easy but I don't know what to do. Is ...
1
vote
2answers
60 views

Tensor Product problem.

Let $V$ be a vector space over $\Bbb F$, and let $x\not=0,y\not=0 $ be two elements in $V$. I want to show that $x\otimes_{_F} y=y\otimes_{_F} x$ iff $x=ay$ where $a\in \Bbb F$. I know the second ...
0
votes
1answer
52 views

field extension $F(x)=F(x^2)$ [duplicate]

Let $x$ be algebraic over $F$ such that the field extension $F(x):F$ satisfies $[F(x):F]$ odd. Then prove $[F(x):F]=[F(x^2):F]$ hence $F(x)=F(x^2)$. How to prove? I only obtained the proof for the ...
1
vote
3answers
428 views

Basis of Vector space $\Bbb C$ over rational numbers.

What will be the basis of vector space $\Bbb C$ over field of rational numbers? I think it will be an infinite basis! I think it will be $B=\{r_1+r_2i \mid r_1, r_2 \in \Bbb Q^{c}\}\cup\{1,i\}$. ...
1
vote
1answer
120 views

Injective but not divisible module

I want to find an injective but not divisible $R$-module. If $R$ is integral domain, every injective is divisible so it should be $R$ is not an integral domain. Is there any example?
2
votes
2answers
83 views

Why does the center of $R$-$\mathbf{mod}$ consist of multiplication maps induced from central elements of $R$?

The center of a category $C$ is the class of natural transformations from $1_C$ to $1_C$. In $R-\textbf{mod}$, I have been able to show that the morphisms $\eta_M(c):x\mapsto cx$ for $x\in M$ and $M$ ...
4
votes
3answers
261 views

similar matrices over $\mathbb{Z}$

Let $A,B$ be $2\times 2$ matrices over $\mathbb{Z}$. Suppose $x^2+x+1$ is the characteristic polynomial for both $A,B$. Determine whether $A,B$ are similar to each other over $\mathbb{Z}$. How to ...
1
vote
1answer
616 views

matrices with same minimal and characteristic polynomials are conjugate [duplicate]

Let $A$, $B$ be two $n\times n$ matrices over $\mathbb{C}$. If $A$, $B$ have the same minimal polynomials and characteristic polynomials, then can we prove that $A$, $B$ are conjugate? i.e., does ...
1
vote
1answer
128 views

Direct product versus direct sum [duplicate]

I want to know the rationale behind and the intuition of why they defined the tensor products. I started to read Keith Conrad's handouts. On page 4, paragraph 2 of this handout he says that the direct ...
1
vote
1answer
89 views

Span of a f.g. $R$-module over the quotient field of $R$.

Let $R$ be an integral domain with quotient field $K$, $R \neq K$. Let $V$ be a finite dimensional vector space over $K$ and $M$ a finitely generated $R$-submodule of $V$. My question is how $KM=V$ is ...
2
votes
1answer
129 views

non degenerate bilinear map for modules

Let $R$ be a commutative ring with 1. Suppose $A,B$ are $R$-modules, $P:A\times B\to R$ is a bilinear map that satisfies the following property: if $P(a,b)=0$ for all $b\in B$, then $a=0$. Then is the ...
1
vote
1answer
62 views

When is a simple tensor equivalent to natural multiplication?

Let $R$ be a commutative integral domain; let $M, N$ be $R$-modules. Let $x_1 \in M$ and let $x_2 \in N$. When does the following equivalence hold? $$x_1 \otimes x_2 = x_1x_2$$ The textbook I'm ...
2
votes
1answer
83 views

Flat modules on Stacks Project.

I have been reading through a bit of the material from the Stacks project, and there is a statement that I cannot make sense of. Lemma 10.36.19(7) states: Let $R$ be a ring. (7) Suppose ...
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votes
0answers
116 views

Why do we study modules? [closed]

Let $R$ be an associative ring. It is usual to study the category of $R$-modules and characterize some rings through treatments of some special modules. For example a ring $R$ is a noehterian if and ...
2
votes
3answers
150 views

$0$-th exterior power, empty product of modules and their tensor product

$\def\finiteprod#1#2{#1_{1}\times#1_{2}\times\dots\times#1_{#2}}$In Lang's algebra he defines the tensor product as a universal object in the category of multilinear maps from $\finiteprod En$ where ...
1
vote
0answers
54 views

Vanishing criterion of pure wedges

Let $R$ be a commutative ring, $M$ some $R$-module, and $m,n \in M$. Is there some criterion when $m \wedge n = 0$ in $\Lambda^2(M)$? There are some sufficient criterions, for example that $m \in ...
1
vote
1answer
2k views

RSA encryption/decryption scheme

I've been having trouble with RSA encryption and decryption schemes (and mods as well) so I would appreciate some help on this question: Find an e and d pair with e < 6 for the integer n = 91 so ...
1
vote
1answer
60 views

Every projective f. g. module is f. p.

I want to show that if $P$ is a finitely generated (f.g.) projective module then $P$ is finitely presented (f.p.).
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votes
3answers
97 views

Free $\mathbb{Z}G$ module

Let $G$ denote a group and $\mathbb{Z}G$ the group ring. Let $F$ be a free $\mathbb{Z}G$ module over a set $X$. Why $F$ is free $\mathbb{Z}$ module over the set $\{gx|x \in X, g \in G\}$? Why is it ...
5
votes
1answer
662 views

A problem about an $R$-module that is both injective and projective.

Let $R$ be a domain that is not a field, and let $M$ be an $R$-module that is both injective and projective. Prove that $M= \left \{ 0 \right \}$. This is exercise 7.52 of Rotman's Advanced ...
0
votes
1answer
63 views

Construction of module on an Abelian group.

Let $M$ be an Abelian group and let $\operatorname{End} M$ be the set of all endomorphims on $M$. Then $(\operatorname{End} M,+,\circ)$ forms a unitary ring. In this setting, if $R$ is a unitary ring ...
0
votes
0answers
49 views

If $M$ satisfies ACC, do we have a similar result?

Let $M$ be an $R$-module and $f$ an $R$-endomorphism on $M$. If $M$ satisfies DCC then There exist $n\in\mathbb{N}$ such that $Imf^{n}+Kerf^{n}=M$. If $M$ satisfies ACC, do we have a similar ...
6
votes
1answer
66 views

$M\times N$ Doesn’t Have a Module Structure

In Keith Conrad's notes (page 4) is written: For two $R-$modules $M$ and $N$ , $M\oplus N$ and $M\times N$ are the same sets, but $M\oplus N$ is an $R-$module and $M\times N$ doesn’t have a ...
3
votes
1answer
201 views

Some questions about Fitting ideals

Let $R$ be a ring and $M$ a finitely presented $R$-module. Given a free presentation $$ R^{\oplus m} \to R^{\oplus n} \to M \to 0 $$ we define $Fitt_k(M)$, the $k$-th Fitting ideal of $M$, to be the ...
1
vote
3answers
2k views

Multiplicative inverse of 97 modulo 386?

A little help would be a lifesaver :) I already used the Euclidean algorithm to find the GCD of $386$ and $97$, which is $1$. However, I'm stuck on this question posed by my professor: "Then use your ...