# Questions tagged [modules]

For questions on modules over rings.

459 questions
Filter by
Sorted by
Tagged with
105 views

128 views

### Injective hull of quotient of modules

Let $R$ be a Noetherian commutative ring. If $N\leq M$ are $R$-modules, then is $E(M)$ isomorphic to a submodule of $E(N)\oplus E(M/N)$? Here $E(M)$ denotes the injective hull of $M$.
263 views

Let $R$ be a ring and $M$ be a $R$-module. Let $rad(M)$ be the radical of $M$, that is, the intersection of all maximal submodules of $M$. Moreover, let $soc(M)$ be the socle of $M$, that is, the sum ...
130 views

516 views

### Dual of a bimodule

For a noncommutative ring $R$, and an $R$-$R$-bimodule $B$, is there a "correct/natural" notion of a dual bimodule? I am interested, really, when $B$ is projective as a left $R$-module. Note: ...
461 views

### Parabolic Kazhdan-Lusztig Conjecture

In this post, Humphreys provides a reference concerning about Kazhdan-Lusztig Conjecture for arbitrary weights in $\mathfrak{h}^*$, which is under the name: Kashiwara and Tanisaki - Characters of ...
173 views

### What are the (co)algebras for the $(\operatorname{Hom}(A,-), A\otimes-)$ adjunction (co)monad?

A module $A$ over a commutative ring $k$ gives a pair of adjoint endofunctors, $(A\otimes_k-)$ left adjoint to $\operatorname{Hom}_k(A,-)$. They produce a monad $T_A$ and a comonad $C_A$. Is there any ...
246 views

### Faithfully flat descent for modules from the simplicial point of view

Let $R \rightarrow R'$ be a faithfully flat ring map, let $M$ be an $R$-module, and let $M_n$ be the base change of $M$ to the tensor product of $n + 1$ copies of $R'$ over $R$. One way to formulate ...
139 views

### Row rank and column rank of matrix with entries in a commutative ring

Let $R$ be a unital commutative ring and $A\in M_{n\times m}(R)$. Under which appropriate invariant "rank" of modules discussed in "Ranks of Modules" one can say that the row rank of $A$ is ...
175 views

### What are all pairs $(R,M)$ of a ring $R$ and a two-sided $R$-module $M$ such that all endomorphisms of $M$ are scalar multiples of $\text{id}_M$?
I was playing with some endomorphism rings and got curious whether there is a classification of all two-sided (not necessarily unitary on any side) modules $M$ over a (not necessarily unital) ring $R$ ...