1
vote
2answers
31 views

$\mathbb{F}_{p}A$-module

Yesterday I was introduced to the definition of a module in a course: "Homological Algebra". I'm doing a project involving $p-$groups and in the text I got from my supervisor they use the word: ...
0
votes
1answer
26 views

Module Notation $Dx$

Let $K$ be a field, and let $D$ be a subring of $K$ with identity. Let $K^*$ be the multiplicative group of nonzero elements of $K$. The group $U$ of units of $D$ is a subgroup of $K^*$. We take ...
3
votes
2answers
104 views

Question about the radical of the Jacobson radical.

I am confused about the notation $\operatorname{rad}^2 A$. It can be considered as $\operatorname{rad}(\operatorname{rad}(A))$ or as $(\operatorname{rad}(A))^2$. Are ...
12
votes
2answers
903 views

Is $A \times B$ the same as $A \oplus B$?

When $A, B$ are $K$-modules, then $A \times B$ is the same as $A \oplus B$. Let $A, B$ be two $K$-algebras, where $K$ is a field. Is $A \times B$ the same as $A \oplus B$? Thank you very much. ...
1
vote
2answers
90 views

Notation for an indecomposable module.

If $V$ is a 21-dimensional indecomposable module for a group algebra $kG$ (21-dimensional when considered as a vector space over $k$), which has a single submodule of dimension 1, what is the most ...
1
vote
1answer
107 views

Notation for the set of all isomorphisms between two modules.

Let $R$ be an arbitrary ring. Given two $R$-modules $A$ and $B$, we may denote the set of all $R$-homomorphisms from $A$ to $B$ by $\operatorname{Hom}_{R}(A,B)$. If in addition we know that $A$ and ...
2
votes
1answer
112 views

Notation for modules.

Given a module $A$ for a group $G$, and a subgroup $P\leq G$ with unipotent radical $U$, I have encountered the notation $[A,U]$ in a paper. Is this a standard module-theoretic notation, and if so, ...
0
votes
1answer
66 views

What exactly is the structure $IM$ for $I$ an ideal and $M$ a module?

If $M$ is an $R$-module, and $I$ is an ideal of $R$, then what is $IM$? That is, does it consist of all products of some form or finite sums of some specific form?
0
votes
1answer
85 views

Superscript wedge on an $R$-module

I'm reading these lecture notes here about the tensor product and the author uses $M^\vee$ where $M$ is an $R$-module. Look at page 2 for an example (example 2.1). What does it mean? I looked through ...