Definition: Call a mapping $f: \mathbb{Z} \rightarrow \mathbb{Z}$ a generalized polynomial if for any distinct integers $m$ and $n$ we have $(m - n)|(f(m)-f(n))$. It is easy to ch …

(Edited) Both notions seem to measure "largeness" of an ideal in a ring. how are they related? does one imply the other and vice versa? i am reading this book "Hereditary Noether …

it seems that a module over a noetherian ring R is finitely generated if and only if it has finite length (sorry, it turns out to be false! i must have had a misunderstanding!) b …

In the category of $\mathbb{Z}$-modules, there exists a module $A$---for instance $\bigoplus_{k=2}^\infty \mathbb{Z}/k\mathbb{Z}$---such that a $\mathbb{Z}$-module $B$ is injective …

Let $R$ be a ring, $\Sigma$ be a multiplicatively closed subset of $R$. $M$ is an $R$-module. Denote the injective hull of $M$ by $E(M)$. $M$ is $\Sigma$-torsion if for any $m$ …

Hy guys. I'm doing some independent analysis which makes use of the tensor product of modules (over commutative rings with unit 1, and ring homomorphisms map $1 \mapsto 1$). Let $\ …

Let $\mathcal{C}$ is a grothendiect category and consider all of what follows in $\mathcal{C}$. Let $${\varepsilon_i: 0\to A_i \to B_i \to C_i\to 0\ ,\ \phi_i^j}$$ be a direct sy …

I have two finite rank free modules $M,N$ over a principal ideal domain (the ring of formal complex power series to be exact) and a homomorphism between these two modules $\phi:M\r …

In Atiyah and Bott's paper "The Moment Map and Equivariant Cohomology", they say that for any exact sequence of modules over $\mathbb{C}[u_1,...,u_l]$ $$D \to E \to F,$$ we have th …

Suppose we know the presentation matrix of a module. (For simplicity: the module is over a local Noetherian ring, the ring is over a field.) For which invariants of the module some …

Let $A=K[[X_1,\dots,X_n]]$ where $K$ is a field. Let $M$ be a finitely generated torsion-free $A$-module, such that for all $k$, the $A[1/X_k]$-module $M[1/X_k]$ is free of rank …

Let $\cal{E}$ be a finitely generated projective bimodule over a (noncommutative) algebra $A$. Moreover, let us assume that $\cal{E}$ is of finite rank $n$. The tensor product $ …

Let $P$ be a pro-p group. Assume that there is a filtration of $P$ by normal subgroups $P_i$ such that $P_0=P$ and $P_{i+1} < P_i(i\in\mathbb N)$. Let $V$ be an $l$-adic represe …

Let $k$ be a commutative ring, a $R$-Bimodule $M$ over a $k$-algebra $R$ is a $k$-module with two actions of $R$ on $M$, on the left and on the right, the classical example of this …

Is there infinite generated reflexive module?