1
vote
0answers
9 views

Module over an associative ring with unity and axioms of projective geometry

According to Wikipedia(http://en.wikipedia.org/wiki/Projective_geometry#Axioms_of_projective_geometry), the axioms of projective geometry due to A. N. Whitehead are: G1: Every line contains at least ...
2
votes
0answers
26 views

Endomorphism rings of a $k$-algebra

Let $k$ be a field and $R$ be the $3$-dimensional $k$-algebra $\left [\begin{array}\ k & k \\ 0 & k \end{array} \right ]$. I have two conjectures: (1) Is any simple right $R$-module has ...
0
votes
1answer
23 views

$H^1(G,A)\rightarrow H^1(H,A)$ is onto

Note that $H^1(G,A)\rightarrow H^1(H,A)^{G/H}$ where $A$ is $G$-module and $H$ is a subgroup in $G$ But I suspect that ${\rm Res}\ : \ H^1(G,A)\rightarrow H^1(H,A)$ may be onto since $ f\in C^n ...
0
votes
1answer
29 views

Proving a submodule is Noetherian

I'm doing some independent study with a professor in Ring/Field theory (I'm an undergraduate) and I have been having a hell of a time wrapping my head around problems involving Noetherian rings and ...
0
votes
0answers
22 views

Projective Resolutions of Finite Type.

Definition. An $R$-module $M$ is said to be of type $FP_n$ if there is a projective resolution $\mathscr{P}$ of $A$ with $P_i$ finitely generated for all $i\leq n$. If the modules $P_i$ are finitely ...
2
votes
1answer
66 views

Self-injective ring but not semisimple?

It is well-known that if $K$ is a field, then $K[x]/(f(x))$ is a self-injective ring for any polynomial $f(x)$ in $K[x]$. On the other hand, we know that a ring $R$ being semisimple is equivalent to ...
1
vote
2answers
27 views

Injectivity of Inflation Homomorphism $H^1(G/K,A^K)\rightarrow H^1(G,A)$

I want to prove that injectivity of $inflation\ homomorphism$ $$0\rightarrow H^1(G/K,A^K)\rightarrow H^1(G,A)$$ where $K$ is normal. Proof : (a) If $x\in A^K$ then $gH\cdot x:= gx$ This is ...
0
votes
0answers
52 views

If $R$ is a integral domain then $TM$ is torsion module

Let $R$ be an integral domain. An $R$-module $M$ is torsion if $TM = M$. Prove that $TM$ is torsion. I'm confused with this exercise, i need to prove that $TTM = TM$ but we know that $TM$ is a ...
0
votes
0answers
22 views

$ A\rightarrow M^G_H(A)$ induces ${\rm Res} : H^n(G,A)\rightarrow H^n(H,A)$

I want to prove the following : $A$ is $G$-module. Then prove that $$ A\rightarrow M^G_H(A)$$ induce a restriction map $$ {\rm Res}\ : H^n(G,A) \rightarrow H^n(G,M^G_H(A)) =_\text{Shapiro} ...
1
vote
1answer
61 views

Complement but not direct summand; Lam, Lectures on Modules and Rings, Example 6.17(5)

Let $S$ be a submodule of an $R$-module $M$. A submodule $C⊆M$ is said to be a complement to $S$ (in $M$) if $C$ is maximal with respect to the property that $C∩S=0$. (This does exist by Zorn Lemma.) ...
7
votes
2answers
109 views

The Freyd-Mitchell Embedding Theorem and projective (injective) objects

Given a small abelian category $\mathcal{A}$, the Freyd-Mitchell Embedding Theorem gives me a fully faithful exact functor $F:\mathcal{A}\rightarrow R$-$\mathsf{Mod}$, for some unital ring $R$, so ...
0
votes
1answer
33 views

How to show $\alpha=i_L$ and other equalities without using isomorphism?

This is a question of Commutative Algebra.It was given in my class.I was able to solve it to some extend.But I have some doubts. Please help me. Thnx in advance. $A$ is a commutative ring with ...
1
vote
1answer
24 views

A uniform module as an intersection

Let $R$ be a semiprime, nonsingular ring with finite Goldie dimension u.dim $R_R$. (Nonsingularity means here that $Z(R_R)=0$, where $Z(R_R)$ is the set of elements $x$ of $R$ with $ann(x)$ is ...
2
votes
0answers
25 views

on basic theory over $\mathbb{Z}_4$-linear codes

I am a bit confused about this example in Huffman and Pless's Fundamentals in Error-Correcting Codes. This can be found in Chapter 12: Let $\mathcal{C}$ be a nonempty subset of $\mathbb{Z}{}^4_4$ ...
0
votes
1answer
21 views

Characterization of Zero in a module

Let $M$ be an $R-module$ over a ring $R$. Obviously, $0 \cdot m = 0, \forall m \in M$. Is it also true that $x \cdot m = 0, \forall m \in M$ if and only if $x = 0$?
2
votes
1answer
27 views

If $I$ and $J$ are distinct maximal ideals of $R$, any module homomorphism $R/I\to R/J$ is zero.

I am not sure how to deal with the following problem: $I$ and $J$ are distinct maximal ideals in a commutative ring $R$ and if $\phi$ is a modules homomorphism from $R$-module $R/I$ to $R$-module ...
0
votes
2answers
44 views

The difference between the free $A$-module $A^{(M\times N)}$, and $M\times N$

Let $M$ and $N$ be two $A$-modules, where $A$ is a commutative ring. What is the difference between the free $A$-module $A^{(M\times N)}$, and $M\times N$? In Atiyah-Macdonald, both of these seem to ...
0
votes
1answer
36 views

Proof of $0\rightarrow A^G\stackrel{f}\rightarrow B^G\stackrel{h}\rightarrow C^G\rightarrow 0$

When we have a $G$-modules exact sequence $0\rightarrow A\stackrel{f}\rightarrow B\stackrel{h}\rightarrow C\rightarrow 0$ we have a $G$-modules exact sequence $0\rightarrow ...
0
votes
1answer
32 views

Can the integral group ring construction be extended to groupoids in such a way that it provides a functor from Groupoids to Rings?

If $G$ is a group, and $R$ is a ring, one can construct the group ring $R[G]$, which is a free $R$-module with basis $G$, and with multiplication coming from the original multiplication of $G$. ...
1
vote
0answers
25 views

What maps of $k$-algebras $A\to B$ induce finite maps $\mathrm{Ext}_B^*(k,k)\to\mathrm{Ext}_A^*(k,k)$?

Let $k$ be an algebraically closed field, and let $A$ and $B$ be finitely generated $k$-algebras. A map $\varphi:A\to B$ of $k$-algebras induces a map ...
1
vote
0answers
38 views

Proof of ${\rm Tor}\ (A,B)={\rm Tor}\ (B,A)$

Problem : I want to prove that ${\rm Tor}$ is symmetric where $1\in R$ Definition : Note that if $B$ has a projective resolution $$ P_n\rightarrow_{d_n}\cdots P_0\rightarrow B\rightarrow 0 $$ By ...
0
votes
1answer
66 views

Proof of the theorem: Any $R$-module is contained in an $R$-injective module

Note that this an exercise in Foote and Dummit's Algebra. Problem: Let $1\in R$. Any $R$-module $M$ is contained in an $R$-injective module. Definition : $R$-module $Q$ is injective if ...
0
votes
1answer
77 views

Does every free $R$-module have a maximal proper submodule?

Let $R$ be a commutative ring with $1$. We know that every finitely generated $R$-module has a maximal proper submodule. Is it true for any free $R$-module? In particular, can we do the following: ...
1
vote
1answer
28 views

Extension of an $R$-homomorphism as a sum

I want to solve this problem: Let $M$ be an injective module over a ring $R$ with identity, and $f$, $g$, $h$ are elements of the ring of $R$-endomorphisms $\operatorname{End}(M)$ such that $\ker ...
-1
votes
3answers
70 views

Annihilator of annihilator of annihilator of a submodule

Let $R$ be a commutative ring and M be an $R$-module. The question is whether for every submodule $N$ of $M$ the equality ...
0
votes
1answer
36 views

A direct limit of pullbacks

Let an $R$-module $C$ be a direct limit of finitely presented $R$-modules $C_i$, and we have a short exact sequence as follows: $$0→A↪B\stackrel{\pi}→C→0.\qquad (S)$$ From each $C_i$ to the direct ...
0
votes
0answers
44 views

Bass numbers of minimax modules are finite?

Let $R$ be a commutative Noetherian ring, and $M$ be a minimax $R$-module. Are the Bass numbers of $M$ are finite? (An $R$-module $M$ is called minimax, if there is a finite submodule $N$ of $M$, such ...
0
votes
1answer
75 views

Modules with finite support in $\mathrm{Max}(R)$

Let $R$ be a commutative Noetherian ring, and $M$ be an $R$-module. Is the following statement true? If $\mathrm{Supp}_R(M)$ (support of $M$) is a finite subset of $\mathrm{Max}(R)$ (the set of all ...
3
votes
1answer
38 views

Injective hull of $\mathbb{ Z}_n$ [duplicate]

What is the injective hull of $\mathbb Z_n$? I know that in case $n=p$ is prime, the injective hull would be isomorphic to $\mathbb Z_{p^∞}$, but in general case, I have no idea. Can anyone be of ...
0
votes
1answer
34 views

Is this ring Noetherian

If $R=\{\frac m n\in{\mathbb{Q}}: \mbox{7 does not divide $n$}\}$. Is $R$ Noetherian? What i think is the ideals of $R$ are finitely generated. But how do i prove this, i have no idea. Can anyone ...
0
votes
1answer
23 views

Submodule Equality

I was wondering whether the following is true or not: Suppose $N$ and $L$ are submodules of $M$ with $N\subseteq L$. If $M/L\cong M/N$ then $L=N$. I am particularly curious about the case when ...
2
votes
1answer
27 views

Infinite dimensional Vector Spaces and Bases of Quotients

Given a basis $\mathcal{B}$ of an infinite dimensional vector space $V$, will a quotient of $V$ by a subspace $I$ necessarily have a basis? Is there a way of obtaining it using $\mathcal{B}$? My ...
0
votes
1answer
62 views

$A\otimes_RB$ is an $R$-algebra

This is Proposition 21 on page 374 of Dummit and Foote's Algebra: If $A,\ B$ are $R$-algebras, that is, they are module which have a center containing $R$, then $A\otimes_R B$ is still so where $R$ ...
1
vote
1answer
25 views

Help understanding statement relating to structure of modules over PIDs

Lemma IV.6.11 of Hungerford's Algebra is Lemma 6.11. Let $R$ be a principal ideal domain. If $r \in R$ factors as $r = p_1^{n_1} \cdots p_k^{n_k}$ with $p_1,\ldots,p_k \in R$ distinct primes and ...
2
votes
0answers
77 views

If $R$ is a domain and $M$ a finitely generated $R$-module, is it true that $\bigcap_{f\in M^{*}}\ker{f}=\operatorname{Tor}M$?

Let $R$ be a domain and $M$ a finitely generated $R$-module. Let $M^{*}=\hom_{R}(M,R)$. Let Tor$M$ be the torsion submodule of $M$. It it true that $$\displaystyle\bigcap_{f\in ...
0
votes
1answer
35 views

$0\to L\to M\to N\to 0 $ is split if $0\to {\rm Hom}_R(D,L)\to {\rm Hom}_R(D,M)\to {\rm Hom}_R(D,N)\to 0$ is exact for any $D$.

Prove that $0\rightarrow L\rightarrow M\stackrel{\phi}\rightarrow N\rightarrow 0 $ is split if $$0\rightarrow {\rm Hom}_R(D,L)\rightarrow {\rm Hom}_R(D,M)\rightarrow {\rm Hom}_R(D,N)\rightarrow 0$$ ...
0
votes
0answers
44 views

Polynomial Ideals and Transverse modules

I have a given ideal and I want to find the "smallest" ideal so that when I add it to the original one, I get another certain ideal. Let $\mathcal{E}$ be the ring of smooth function germs ...
0
votes
2answers
56 views

Existence of module homomorphism $\phi : M \rightarrow N$ such that $\Phi =\phi^2$

This is a homework problem and I have solved part (a), but I am not sure how to approach part (b). Should I approach it the way how the universal property of free modules are defined? Any hint(s) ...
-1
votes
1answer
62 views

How does the free abelian group of $M \times N $ have $M\times N$ as basis in construction of tensor product of modules?

With M and N being R-modules, how does Z(M,N) have $M\times N$as a basis and therefore becomes a free abelian group? Consider the element n(m,0) for an element$\,m\in M $ of order n. This is zero ...
1
vote
0answers
52 views

Direct proof that $\mathbb{Z}/p\mathbb{Z}$ is not a flat $\mathbb{Z}/p^2\mathbb{Z}$-module.

I am trying to prove that $\mathbb{Z}/p\mathbb{Z}$ is not a flat $\mathbb{Z}/p\mathbb{Z}$-module. The reasoning I have is the following. We have an exact sequence $0 \to \mathbb{Z}/p\mathbb{Z} ...
1
vote
0answers
26 views

Example of a ring of finite global dimension with flat qu0tient

I've been thinking about this for quite a while but I cannot seem to find an example of If $k$ is a commutative ring of finite global dimension and I'm looking for a strictly not-commutative ...
2
votes
0answers
47 views

Structure theorem of modules in the graded case

I ran into an exercise in D. Passman's book A course in ring theory (page 130) asking me to prove the structure theorem of modules over PIDs in the graded case. Could anyone point me in the right ...
2
votes
3answers
62 views

Contrasting definitions of bimodules? An illusion?

Recently my definition of a bimodule over a $k$-algebra has been challenged and I believe both definitions to be equivalent, am I wrong? Notation: $k$ is a commutative ring and $A$ is a (unital ...
-1
votes
1answer
71 views

Any module is the colimit of its finitely generated submodules.

Fact A : If $E_i$ is a directed system of $R$-modules, and $F$ is another $R$-module. Then $$\varinjlim(F\otimes E_{i})=F\otimes(\varinjlim E_{i}).$$ Fact B : "Every module is a direct limit of its ...
1
vote
1answer
41 views

Question about Universal Mapping Property

Now I seem to understand the construction of the tensor product of $R$-Modules by defining $$F_R(M \times N) := \bigoplus_{(m,n) \in M \times N} R\delta_{(m,n)}$$ and constructing the submodule $D$ in ...
1
vote
0answers
61 views

Tensor product of free modules over free algebra

Suppose $M$ and $N$ are modules over a (commutative, unital) ring $S$. Let $R$ be a subring of $S$ such that $S,M$ and $N$ are all free, finitely generated modules over $R$. Question: Under what ...
2
votes
0answers
47 views

How are 'irreducible' elements called in module theory?

Let $R$ be a 'nice' ring (whatever that means, I guess commutative with unity $1_R$) and let $M$ be an $R$--module. Is there a name for the following property? An element $m \in M$ is called ... iff. ...
0
votes
2answers
76 views

Graded rings and modules - which book to refer to?

Could you recommend a book with a good treatment of the theory of graded rings and modules?
3
votes
2answers
65 views

module over a quotient of a principal ideal domain

The Statement I suspect the following proposition is well known, but I found no reference. Proposition If $A$ is a principal ideal domain, if $I$ is a nonzero ideal of $A$, and if $M$ is an ...
1
vote
1answer
36 views

Product of divisible module is divisible

I have the following problem Is the product of divisible $R$-modules divisible? I think it is not. But I need some counterexample to this, somebody can give me a "place" to search one? Thanks a ...