The modules tag has no usage guidance.

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### Constructing a simple $A$-module

Let $n \ge 2$, and let $A$ be the (unital and associative, but noncommutative) $\mathbb{C}$-algebra with generators $x_1, \dots, x_n$ and relations $x_ix_j + x_j x_i = 2\delta_{ij}$. What is a ...

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### Cyclic faithfully flat modules

Iam looking for an example of a cyclic faithfully flat R-module but not projective. Could someone help me?

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### trying to understand particular modules, to contruct some nice free resolutions

Let $R$ be a local or graded ring, fix some morphism $\phi\in Hom_R (R^{\oplus n},R^{\oplus m})=:Mat(m,n;R)$.
In various applications one meets modules like $\frac{Mat(m,n;R)}{Span(U ...

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### When is a polynomial ring free over a graded subalgebra?

Keep the setting of my previous question and let $I := k[x_1, \dots, x_n] \cdot A_{>0}$ be an ideal of the algebra $k[x_1, \dots, x_n]$ generated by the set $A_{>0}$. It is clear that $I$ is a ...

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### Showing that the stable module category of a ring $R$ restricted to maximal Cohen-Macaulay objects is trivial if $\text{gldim } R < \infty$

(In the following, a (not necessarily commutative) ring $R$ is Gorenstein if it has finite injective dimension as a module over itself on either side, and a finitely generated (right) $R$-module is ...

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**1**answer

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### Projectivity of torsion-free modules over integral group rings

Let $G$ be a torsion-free group and assume that the integral group ring $\mathbb{Z}G$ is torsion-free as well. Let $M$ be a torsion-free, finitely generated module over $\mathbb{Z}G$.
If we assume ...

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136 views

### locally noetherian categories and the category of quasi-coherent sheaves over a noetherian scheme

It is known that a ring $R$ is noetherian if and only if direct sums of injective $R$-modules
are injective if and only if every injective $R$-module is a direct sum of indecomposable injective ...

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55 views

### Semisimple monoidal category with duals

We say that an abelian category is semisimple if every object is a semisimple object, which is to say, a direct sum of finitely many simple objects.
Let $({\cal C},\otimes,*)$ be a semisimple ...

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### some sort of 'saturation' of module quotients

Let $R$ be a local Noetherian ring over a field, with the maximal ideal $\mathfrak{m}$. (e.g. $R=k[[x_1,\dots,x_{p>1}]]$) Given two $R$-modules, $N\subset M$, of the same (finite, non-zero) rank. ...

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184 views

### In modules over finite rings $Rv = Rw \iff R^\times v = R^\times w$

Let $R$ be a finite ring (with unit, possibly non-commutative), and $M$ a left module over $R$. Let $v,w\in M$. Then
$$Rv = Rw \iff R^\times v = R^\times w.$$
This follows from Lemma 6.4 in ...

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109 views

### Complex plane mod lattice to elliptic curve correspondence generalization

If we observe the correspondence
$$\mathbb{C}/\Lambda \rightarrow E: Y^{2} = X^{3} - \frac{g_{2}(\Lambda)}{4}X - \frac{g_{3}(\Lambda)}{4},$$
we see the relationship between weight 4 and weight 6 ...

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93 views

### If the direct sum of cyclic modules is cyclic, what happens to nontrivial extensions?

Let $R$ be any ring with unit over some field $k$ and let $M_1$ and $M_2$ be cyclic left $R$-modules with $dim_k(Ext^1_R(M_2,M_1))\geq 1$.
Assume $M_1\oplus M_2$ is a cyclic left $R$-module. Given ...

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131 views

### M is an R-module which is not finitely generated. is it true that $\inf \{ i| H^i_I(M)\neq 0 \}\le ht_M I?$

Let $R$ be a commutative noetherian ring (with $1$), $I$ an ideal of $R$, and $M$, a finitely generated $R$-module such that $IM \neq M$. Then, by Theorem 6.2.7 of BRODMANN-SHARP's Local Cohomology ...

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### Are finitely presented modules finitely presentable? [closed]

Over a ring $R$ we have a notion of finitely presented module, namely:
Definition 1 A module $F$ is finitely presented if there are $m$, $n$ positive integers such that $R^m\to R^n\to F\to 0$ is ...

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156 views

### Regularity of a tensor product

Let $A \subseteq B$ and $A \subseteq C$ be commutative noetherian domains.
Assume that $A$ and $C$ are regular rings (=every localization at a maximal ideal is a regular local ring).
Assume that $B$ ...

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**1**answer

141 views

### When every module is a scalar extension?

Let $A \subseteq B$ be commutative noetherian domains.
Of course, if $M$ is an $A$-module, then $M \otimes_A B$ is a $B$-module.
I am curious to know if there exist additional conditions on $A$ and ...

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60 views

### Morphism of modules of sections and pullback bundles

I'v asked this question on StackExchange but unfortunately nobody answered. I thought that maybe it would be more apropriate to post it here:
so suppose that we have a morphism $\theta: \Gamma(B,E_1) ...

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### behavior of multiplicity in exact sequences

Bruns-Herzog define multiplicity When the ring and module are not necessarily graded as $e(M)=e(gr_m(M))$, see B-H 4.6. I have two questions:
Question1. Many concepts in commutative algebra have ...

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### Derived categories of modules categories

Does anyone know if there is a note or a paper about the derived category of the category $\sigma[M]$ where $M$ is a left module over a ring?, and some uses of this.

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133 views

### When are countably generated Hilbert modules generated by c.p.c. order zero maps?

Throughout let $B$ be a stable C*-algebra, i.e. $B\cong B\otimes K$, where $K$ is the C*-algebra of compact operators on an infinite dimensional separable Hilbert space. It is well-known that any ...

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### Commutation of tensor products with inverse limits in a specific case

For $X,Y$ sets, let's denote $Y^X$ the set of all mappings $X\rightarrow Y$. If $Y(=R)$ is a ring, $R^X$ is a $R$-module (well, a bi-module but my question is - at first - concerning commutative ...

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### When does a faithful module have an element with zero annihilator?

This is a follow up of Example of a finitely generated faithful torsion module over a commutative ring.
Let $M$ be a finitely generated module over a commutative ring $R$ with the property that ...

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### Is this apushout diagram [closed]

Let $A, B, C, E$ and $F$ be some objects in an abeleian category $\mathcal{C}$. Let we have a commutative diagram
\begin{array}{ccccccccc}
0 & \xrightarrow{} & A & \xrightarrow{f} & ...

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### A noncommutative vector bundle associated with a codimension one foliation

Assume that we have a codimension one foliation of a manifold $M$ which is generated by a one form $\alpha$. So the following $\phi$ satisfies $\phi \circ \phi =0$:$$\phi:\Omega^{i}(M)\to ...

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134 views

### Annihilator of tensor product when $R$ is domain

Let $R$ be a Noetherian domain and $M$ and $N$ be two faithful $R$-modules. Is it true that $\operatorname{Ann}_R(M\otimes_R N)=0$?

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### $IM=mM$. can we say that $I$ is a reduction ideal of $m$?

Question. Let $(R,m)$ be a Noetherian local ring and $M$ be a finite faithful $R$-module. Let $I$ be an ideal of $R$ such that $IM=mM$. Can we say that $I$ is a reduction ideal of $m$? Recall that $I$ ...

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### canonical module can be identified with an ideal. how can one reach that ideal?

Let $R[[X,Y,Z]]/(X,Y)\cap (Y,Z)\cap(X,Z)$. then $R$ is Cohen-Macaulay ring and has a canonical
module, $K$. By Proposition 3.3.18 of Bruns_Herzog, $K$ can be identified with an ideal in $R$.
So we ...

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### Graded version of Baer's Criterion

Baer's Criterion for injectiveness of modules says: "An $R$-module $E$ is injective iff for all ideals $I$ of $R$, every homomorphism $f\colon I \to E$ can be extended to $R$." I wonder if there is a ...

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225 views

### When are homotopy categories of model categories closed modules over the homotopy category of $(\infty, 1)$-categories?

Let $\mathrm{Quillen}$ be the model category of simplicial sets with the Quillen model structure, and $\mathrm{Joyal}$ the model category of simplicial sets with the Joyal model structure.
As is ...

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### Homomorphisms from projective modules

Let $B$ be a $A$-algebra which is free of finite rank as $A$-module. Let $X$ be a finitely generated projective left $B$ module. (So $X$ is also a f.g. projective $A$ module.) Are these homomorphism ...

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### An embedding of modules by tensor product over a Noetherian domain

I have a problem on Ring theory. I would like to prove or disprove the following statement:
Let $R$ be a Noetherian domain. Then by the Goldie theorem $R$ have $Q$ as a full ring of quotients and $Q$ ...

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### Multiplicity of $Ext^{d-t}(M,\omega_R)$, ($d=\dim R, t=\dim M$)

Let $R=\bigoplus_{i \geq 0} R_i$ be a Cohen-Macaulay graded ring ($R_0$ is a field and $R$ is generated by $R_1$) of dimension $d$ with canonical module $\omega_R$, and $M$ a graded Cohen-Macaulay ...

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### Existence of small projective dimensioned modules

Suppose $A$ is a (if necessary unital) associative ring and $I$ is a left ideal in $A$. Let $\operatorname{pd}(M)$ denote the projective dimension of a left $A$-module $M$.
Then do either of the ...

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### Correspondences as generalized morphism between $C^*$-algebras

While reading about Morita equivalence in the category of $C^*$-algebras I met also the following notion: a correspondence between two $C^*$-algebras $A,B$ is a pair $(X,\varphi)$ where $X$ is a ...

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### Isomorphism of Hilbert modules

In Hilbert space theory if we have a linear bijection $U:H \to K$ which is an isometry, then it preserves inner products. Suppose now that you have two (right) Hilbert modules $X,Y$ (say, over the ...

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### Ext functor for more than two modules? [closed]

The question is natural. Let's just work in the category of modules over a ring. Pick three modules $M_1, M_2, M_3$. Consider consecutive extensions of these modules, i.e., consider M, such that we ...

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260 views

### Example of a Frobenius algebra that is not projective over a Frobenius subalgebra

I'd like to know an example of a Frobenius algebra $A$, with a subalgebra $B$ that is itself a Frobenius algebra, such that $A$ is not projective as a left $B$-module. I don't require any ...

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### what are the possible approximations for ideals

(Fix some local ring $(R,\mathfrak{m})$ over a field of zero characteristic.)
Suppose an ideal $J$ is defined by some complicated formula/procedure. And there is no hope of computing it/or writing ...

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### Reference request for stably free modules

I am looking for some references on the theory of stably free modules. I will call (F) the following property for a ring $R$: every f.g. stably free module over $R$ is free.
1) Is there a standard ...

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237 views

### Uniform Artin-Rees

The Artin-Rees lemma states that if $R$ is a Noetherian ring, $I \subseteq R$ is an ideal and $N \subseteq M$ are finitely generated $R$-modules, then there exists an integer $k$ such that for every ...

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### Lifting a direct summand of a free module

[EDIT]: After getting a nice counter example provided by Steven Landsburg I realize that I forgot to impose an important condition...namely $R$ is supposed to be complete w.r.t. the $I$-adic topology. ...

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### How to generalize balanced and absorbing sets to R-modules?

I'm looking for generalizing the notions of balanced set and absorbing set. The goal is using them for analyzing topological R-modules with R being a unit ring.
It's easy to generalize balanced and ...

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### Let $A\in\operatorname{M}_n(F)$ be a matrix, how to prove $\bigcap_{X\in C(A)}C(X)=F[A]=\frac{F[x]}{(m_A(x))}$

I asked the following question on Math Stack Exchange, but no people reply. I know MO is more professional and it is for mathematicians to discuss research problems. Maybe this question is unsuitable ...

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### a generalization of the annihilator of cokernel ideal

Let $R$ be a (commutative, associative, unital) ring, consider a homomorphism of some (finitely generated) free $R$-modules $F\stackrel{A}{\rightarrow}G$. Its basic invariants are the Fitting ideals, ...

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### Is $\mathbb{R}$ a $\mathbb{C}$-module without AC?

Assuming ZFC. We can make $(\mathbb{R},+)$ into a nontrivial(scaler multiplication is not identicaly zero) $\mathbb{C}$-module.
Now my questions are?
0.Is it consistent with $ZF$ that $\mathbb{R}$ is ...

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### Canonical presentation of pro-modules over pro-rings

Let $A = (\dotsc \twoheadrightarrow A_2 \twoheadrightarrow A_1 \twoheadrightarrow A_0)$ be a (commutative) pro-ring with surjective transition maps. Consider the category $\mathcal{M} := \varprojlim_i ...

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### Ring of endomorphisms as a criterion of a dimension 1 module

Let $R$ be a unital ring and $M$ be an $R$-module. I have some questions about relation between the ring $\operatorname{End}_R M$ of endomorphisms and the notion of “dimension” of a module. I’m not ...

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### Submodule embeddable in a finitely generated module

I have a terminology question. For a commutative ring $A$ (not necessarily Noetherian), $A$-modules that are isomorphic to an $A$-submodule of a finitely generated $A$-module form a fairly good class ...

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240 views

### Does $ \text{mult}(R / I) = d_{1} \cdots d_{r} $ imply that $ (f_{1},\ldots,f_{r}) $ is an $ R $-regular sequence?

We define the multiplicity of an $ R $-module $ M $ of dimension $ d > 0 $ to be
$$
\text{mult}(M) \stackrel{\text{df}}{=} \text{LC}(P_{M}) \cdot (d - 1)!,
$$
where $ P_{M} $ denotes the Hilbert ...

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### Is a projective module of constant finite rank finitely generated?

If $R$ is a (commutative) ring and $P$ is a projective $R$-module, then every localization of $P$ at a prime of $R$ is free by Kaplansky's theorem, and has a well-defined rank. If these ranks are all ...