0
votes
0answers
22 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 ? ...
0
votes
0answers
30 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 ...
2
votes
1answer
77 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 ...
3
votes
0answers
57 views

Equations in the semiring of f.g. modules

Let $R$ be a commutative ring. Then we may consider the semiring $G(R)$ of isomorphism classes of finitely generated $R$-modules with $+ = $ direct sum, $* = $ tensor product, $0 = $ zero module, $1 = ...
2
votes
0answers
59 views

Question about definition of some classes of bimodules.

Suppose that we have a ring $R$ and a $R$-$R$ bimodule $M$ such that: For every $r\in R$ and $m\in M$ there exists $r'\in R$ such that $m\cdot r=r'\bullet m.$ Examples of this bimodules can be seen ...
0
votes
0answers
42 views

Ring $R$ as $R[x]$-module

My professor mentioned some interesting examples of modules, giving as an example the following two: $R$ as an $R[x]$-module, in which multiplication by $x$ was taken to be evaluation under a fixed ...
2
votes
1answer
34 views

Looking for a concise treatise on the $\mathbb Z^n$ as a $\mathbb Z$-module

I am working on something which turned out to be isomorphic to the $\mathbb Z$-module $\mathbb Z^n$ and want to efficiently learn about its properties without needing to deal too much with more ...
3
votes
2answers
104 views

Module of $R$-valued functions on an infinite set is not countably generated

Let $R$ be an integral domain and $X$ be an infinite set. Let $R^X$ be the set of all functions $f: X \rightarrow R$, viewed as an $R$-module in the usual manner: for $\alpha \in R$, $\alpha f: x \in ...
3
votes
0answers
49 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) ...
2
votes
2answers
64 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?
3
votes
0answers
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 ...
3
votes
1answer
103 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 ...
3
votes
0answers
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 ...
2
votes
1answer
149 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 ...
1
vote
1answer
67 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 ...
1
vote
2answers
86 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.
2
votes
1answer
51 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 ...
2
votes
1answer
111 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 ...
2
votes
1answer
66 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 ...
1
vote
1answer
58 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?
3
votes
0answers
69 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 ...
1
vote
0answers
45 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 ...
1
vote
1answer
133 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 ...
4
votes
0answers
26 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 \begin{equation} \Lambda_R^n(M)\cong ...
7
votes
1answer
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 ...
6
votes
2answers
340 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 ...
14
votes
1answer
434 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 ...
19
votes
2answers
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 ...
2
votes
1answer
275 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 $\lim\limits_{\longrightarrow}\mathrm{Hom}(N_j,M)$ ...
3
votes
3answers
159 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 ...
3
votes
1answer
153 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 ...
2
votes
1answer
83 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 ...
3
votes
1answer
118 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 ...
3
votes
3answers
111 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 ...
4
votes
2answers
288 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 ...
4
votes
2answers
149 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 ...
4
votes
2answers
670 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 ...
7
votes
2answers
342 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 ...
4
votes
6answers
835 views

Some basic book to start with modules?

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 ...
4
votes
1answer
231 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. ...
2
votes
1answer
275 views

Name of a book with the following contents?

Some time ago, I received Algebra notes from my advisor who is advising me on a project. I learned very much from these notes and wondered what the name of the book from which these notes were ...
5
votes
0answers
162 views

Dual modules and first cohomology

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 ...