# Tagged Questions

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 ...
23 views

### Submodule of a free module over a PID with infinite rank [duplicate]

Let $R$ be a PID. We know that a submodule of a free $R$-module with rank $n < +\infty$ is free with rank $\leq n$. But if $M$ is a free $R$-module of infinte rank does this fact remain true ? ...
34 views

### Suggest a good book or reference on graded modules over polynomial rings

I am looking for reference books or papers on graded modules over the polynomial ring $k[x_0, \ldots, x_n]$. Any good commutative algebra text like Eisenbud's Commutative Algebra already contains a ...
84 views

### Quasi-hereditary algebras

Can anyone please recommend a reference (I prefer a book chapter) on Quasi-hereditary algebras? and if it is possible tell me how to prove that Schur algebras are Quasi-hereditary (or any other ...
59 views

52 views

### Endomorphism rings of MCM Modules

Let $k$ be a field (algebraically closed of characteristic not equal to two, if you like) and let $R = k[[t^2, t^{2n+1}]]$. It is well known $R$ has finite type and the MCM (maximal Cohen-Macaulay) ...
67 views

### when is a ring a free module over a subring?

Let $S \subset R$ be rings, $S$ not necessarily an ideal of $R$, and $S \neq R$. Is there anything that can be said about when $R$ is free as an $S$-module?
51 views

### “Localization” of a module at a family of elements

Let $x=(x_i)_{i \in I}$ be a family of elements of a commutative ring $R$. Typically $I$ is infinite. Let $M$ be an $R$-module. For every finite subset $E \subseteq I$ define $M_E = M$, and for ...
110 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 ...
60 views

### Morita theory for simplicial rings

My question is the following: is there an analog of Morita theorem in the simplicial setting? I mean, we can define two simplicial rings $A,B$ to be simplicially Morita equivalent is the categories ...
159 views

### Finding Idempotents?

I was wondering if there is a method for finding primitive idempotents of a finite dimensional algebra (over a field)? or in other words is there any way to build the complete set of primitive ...
68 views

### Module of power series and change of rings

Let $R$ be a commutative ring with unit an $M$ an $R$-module. Define the formal power series over $M$ as $$M[[X]]=\{\sum_{n\ge 0}m_nX^n: m_n\in M\}$$ and make this an $R[[X]]$-module by ...
88 views

### Definition Of 'Inner Derivation'

I need a useful definition of an inner derivation of modules or even better a good reference to read about.
52 views

### characterization of certain idealizers of submodules

Consider two commutative rings with identity, $R$ and $S$ where $R$ is a principal ideal domain, $S$ is free and finitely-generated as an $R$-module and suppose there is a $R$-module homomorphism ...
126 views

### Reference Request: video lecture on module theory

Is there a good online series of introductory video - lectures on module theory? I really like the Abstract Algebra course by Benedict Gross, unfortunately he does not cover modules, hence I was ...
69 views

### Existence of injective hull

Is that true that every module over a ring has an injective hull? The term "has an injective hull" appears in several different contexts, some of them say that modules over a ring has this property ...
60 views

### Reference request for Modular representation theory

I am trying to learn modular representation theory. I would be thankful if any one tell me a good reference to start with?
70 views

### Looking for a short book on modules

I'm looking for a short (100-150 pages) introductory book devoted to the algebra of modules. (I realize that many comprehensive algebra books include coverage of modules, but I'd prefer to read a ...
46 views

### Countably generated modules with local endomorphism rings

I have to give a lecture for undergraduate students about a theorem by Crawley-Jonsson-Warfield which states that if a module $M$ is a direct sum of countably generated modules, each with local ...
140 views

### What's stronger: projective or locally free? flat or locally free?

maybe that's an idiot question, however I did not found anything related in the classical references. It's know that a finitely generated projective $A$-module $M$ is locally free ,since each ...
27 views

### An explicit $\Lambda_R^\ell(M)$ when $M$ is not free

Let $\Lambda_R^n(M)$ be the nth exterior power of an $R$-module $M$. Let us assume $M$ is finitely generated. When $M$ if free, say, $M=R^{\oplus d}$, we know \Lambda_R^n(M)\cong ...
95 views

### Can we really understand $R$ by studying $R$-modules? [duplicate]

According to Algebra: Chapter 0, the category $R\operatorname{-Mod}$ reveals a lot about $R$. However after completing the first eight chapters, I still found no examples where this happens. Can ...
373 views

### Counterexamples in algebra

I got the feeling that whenever a subject gets so sophisticated that Zorn's lemma is needed, a book of counterexamples in that subject would probably benefit researchers/ students a lot. Zorn's ...
475 views

### Alternative construction of the tensor product (or: pass this secret)

The paper Tensor products and bimorphisms by B. Banachewski and E. Nelson studies tensor products (defined by classifying bimorphisms) in concrete categories. It is quite interesting that their main ...
1k views

### Applications of the Jordan-Hölder Theorem.

All books about group theory bring the Jordan-Hölder theorem, but does not give applications. I only saw in Rotman's book the Fundamental Theorem of Arithmetic being proved as a consequence of this ...
300 views

### Direct limits and $\rm Hom$

I read that $\lim\limits_{\longleftarrow}\mathrm{Hom}(N_j,M)\cong\mathrm{Hom}(\lim\limits_{\longrightarrow}N_j,M)$. I was wondering if we can write $\lim\limits_{\longrightarrow}\mathrm{Hom}(N_j,M)$ ...
166 views

### Constructing a basis for finite abelian groups

Let $G$ be a finite abelian group, and $g_1, \ldots, g_k$ a set of "linearly independent elements", namely such that $\langle g_1 \rangle \oplus \ldots \oplus\langle g_k \rangle$. I would like to ...
155 views

### Linear algebra of finite abelian groups

Let $\phi:G \to H$ be a surjective homomorphism of finite abelian groups, and let $g_1, \ldots, g_n$ be an irredundant set of generators (from now on, a basis) for $G$. be a basis for $G$, meaning a ...
85 views

### Reference on Operators on Modules over PID

The operators on finite dimensional vector spaces over any field can be seen in nice form w.r.t. some choice of basis such as "diagonal, triangular, Jordan form, rational form etc." But, for ...
121 views

### The Frobenius-Nakayama Formula

I am currently reading a paper where it refers to the usual Frobenius-Nakayama formula describing quotients of an induced module. It is refering to the following result: If $k$ is a field, $P$ is ...
112 views

### Are the $\mathbb{Z}/p^k\mathbb{Z}$ simple modules all isomorphic to one another?

After looking back over some finite field theory, I've been thinking about the ring $\mathbb{Z}/p^k\mathbb{Z}$ for some prime $p$. I'm just curious, are the $\mathbb{Z}/p^k\mathbb{Z}$ simple modules ...
306 views

### Rewriting automorphism of matrix algebra in terms of automorphisms of the underlying ring?

I've used the following idea as a black box for some time now, but it occurred to me I don't fully understand why it's true. Suppose $A=M_n(R)$ is the algebra of square matrices over some division ...
152 views

### Does totally flat commutative ring imply all ideals are idempotent?

From reading Atiyah and MacDonald, I know of the result that a absolutely flat commutative ring has all principal ideals idempotent. Reading around on math reference, I think that if a commutative ...
700 views

### Classification of simple modules for algebra of upper triangular matrices?

I've been refreshing my linear algebra, and this is a question of curiosity I have. Let $U:=U_n(F)$ be the algebra of upper triangular $n\times n$ matrices over a field $F$. Is there a classification ...
357 views

### Ideal as a projective module

I am sorry, this may not be a good question here. I am looking a good reference about when the ideal $I$ of a given commutative ring $R$ (local or may not be local) with identity is a projective ...
888 views

I am very interested in studying modules, which I have studied algebra, is basic theory of groups and rings, of Hungerford and Dummit. I was reading the Dummit the module, but I still struggled a bit ...
236 views

### Lifting maps of quotient modules

Today I tried to check this, but couldn't see how to do it. I think it is probably a standard result, but a brief check of Atiyah-Macdonald didn't yield anything, and I don't know what to google for. ...
Let $G$ be a finite group, $K$ a characteristic-$p$ algebraically closed field (say $p$ divides $|G|$), and let $M$ be a finite-dimensional $KG$-module. What hypotheses are needed on $G$, $M$ to ...