# Tagged Questions

For questions on modules over rings.

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### On finite Uniform (Goldie) dimensions

1) Is there any characterization of $\Bbb{Z}$-modules with finite uniform dimensions? 2) Find two $\Bbb{Z}$-modules $N, M$ such that $N\leq M$ and $M\hookrightarrow N$ but $N$ is not isomorphic to ...
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### Representation theorem for modular lattices?

Birkhoff's representation theorem implies that every distributive lattice embeds into the lattice of subsets of a set. Is there also some representation theorem for modular lattices? For example, I ...
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### on the generic" modules of finite length (skyscrapers)

Let $R$ be a local or graded ring. (If it helps, can assume the ring is "good", e.g. $R=k[[x_1,..,x_p]]$, where $k$ is a field of zero characteristic.) Let $M$ be a finitely generated $R$-module ...
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### Dimensions of dual vector spaces

Let $V_F$ be an infinite dimensional right $F$-vector space (over a field $F$, or even over a division ring). The dual space $V^{\ast}={\rm Hom}(V,F)$ is naturally a left $F$-vector space (coming ...
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### Analogue of Bass's Lemma 2.4 on when inverse images of free modules are free

Let $R$ be a Noetherian integral domain. Let $x\in R$ be a prime element. Let $\overline{R}=R/Rx$. Let $P$ be a finitely-generated projective $R$-module. Assume that $\frac{P}{xP}$ is a free ...
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### Does an hereditary scalar extension indicate the original algebra is hereditary?

Let $A$ be a finite-dimensional algebra over a field $\mathbb{F}$ of characteristic zero and let $\mathbb{K}/\mathbb{F}$ be a finite galois extension. Assume we know that ...
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### grade of ideals in non-noetherian rings

Let $R$ be a commutative ring with unity, and $M$ an $R$-module. Assume that $I$ and $J$ are finitely generated ideals and $K$ another ideal of $R$. Let $\textbf{x}$ be a sequence of generators of ...
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### Functoriality of $\mathsf{Cu}$

I have always been happy with the proof of the functoriality of the Cuntz semigroup $\mathsf{Cu}$ given in arXiv:0902.3381, where the isomorphism $$\mathsf{Cu}(A)\cong W(A\otimes K)$$ is used, $A$ ...
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### Reference request for $R$-index

Let $R$ be a noetherian domain with field of fractions $F$, let $V$ be a finite-dimensional $F$-vector space, and let $M,N \subseteq V$ be $R$-lattices in $V$ (finitely generated $R$-submodules of $V$ ...
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### Simple bimodule over matrix ring [closed]

Let given not trivial simple $R$- $R$ bimodule $M$, where $R$ - $n\times n$ matrix algebra over field $\mathbf{F}$. Is it true that $M$ is uniquely defined?
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### The number of submodules of $\mathbb{Z}_q^n$

Observe $\mathbb{Z}_q^n = \mathbb{Z}_q \times \cdots \times\mathbb{Z}_q$ as a module over $\mathbb{Z}_q\equiv\mathbb{Z}/q\mathbb{Z}$, for general $q$. I am interested in the following questions: How ...
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### $A$-module is free if and only if equation involving Hilbert-Poincaré series holds, $M$ infinitely generated case

See my question here. Let $A = \oplus_{i \ge 0} A_i$ be a nonnegatively graded commutative algebra and $M$ a nonnegatively graded $A$-module. Assume in addition that $A_0 = k$ and all vector ...
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### Rings all of whose torsion modules are cyclic

Let us call a (possibly non-commutative) ring $R$ "very good" if every finitely generated torsion left $R$-module is cyclic. Here is an example of such a ring: Let $k=\mathbb{C}((t))$ and let ...
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### Quotient modules of polynomial rings by maximal one sided ideal

Let R[X] be a ring of polynomials over an associative unital ring R which is not necessarily commutative. Let M be a maximal left ideal in R[X]. It is easy to see that if the intersection of M with ...
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### Rank versus free-rank of a module

Suppose M is a finitelly generated left module over a ring R. We define the rank of M as the minimal number of generators of M. If in addition M is free, then we define the free-rank of M as minimal ...
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### Irreducible representations of quantum affine algebras

The finite dimensional simple modules of quantum affine algebras are parameterized by Drinfeld polynomials. Some other modules of quantum affine algebras are studied in the paper. Some infinite ...
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### Is this notion of 'closed subset' of a semigroup action something people have thought of?

Suppose $S$ is a semigroup (or a monoid, or a category), and $X$ is an $S$-set -- that is, a set with an action of $S$. Say that a sub-$S$-set $Y$ is "downward closed" (or maybe "well-generated") if ...
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### Teaching an abstract algebra class involving modules, best way to introduce operations on modules?

I am about to teach a few classes on modules and their operations, namely the following: direct product, direct sum, and finitely generated modules. The following is the "skeleton" version of how I ...
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### The use of modules in control theory

So far I have seen the use of vector spaces in control theory and other notions from linear algebra; So I wonder if there's a use of this abstraction of modules over rings in control theory? any ...
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### Jordan-Hölder-like statements for modules with $\Delta$-filtrations over a quasihereditary algebra

Definitions Let $A$ be an Artin algebra (for instance, take $A$ to be a finite dimensional algebra over some field) and label the isomorphism classes of simple $A$-modules by the elements of a ...
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### 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 ...
### 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 ...
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 ...