Let $R$ be a Noetherian ring and $M,N$ finitely generated modules. I want to show that $$\mathrm{Ass}_R(\mathrm{Hom}_R(M,N)) = \mathrm{Ass}_R(N) \cap \mathrm{Supp}(M)$$ We don't …

Nota bene: all rings are supposed commutative with $1$ and all ring homomorphism should be unital. Let $B$ be the set of truth values {True, False}. For a formula $\phi$ denote by …

This looks extremely easy, but then again it's late at night... Let $k$ be a commutative ring with unity. An element $a$ of a $k$-algebra $A$ is said to be transcendental over $k$ …

Hello all, The lemma 3.3 page 214 in Hartshorne Algebraic Geometry book states:" If $I$ is an injective module over a Noetherian ring $A$. Then for any $f\in A$, the natural map o …

Let $A \to B$ be a finitely generated homomorphism between two commutative noetherian rings. As far as I understand, in various generalizations of this situation, such a map is ca …

What is the relation between $H^i_I(-)$ and $H^i_J(-)$ (cohomological functors) when $I\subset J$ are ideals of a (local) noetherian ring?

I'm looking for examples† of rings with the property that for each $P={\rm Ann}_R(a)\in{\rm Min}(R)$ then $a\in R$ is idempotent (ie $a^2=a$) † other than domains!

Suppose that $R$ is a (local) ring and $r\in R$. When do the equations $Ann_R(Ann_R(r))=Rr$ or $\sqrt{Ann_R(Ann_R(r))}=\sqrt{Rr}$ hold? I already know that it holds for Artinian G …

I'm looking for an example of a commutative (preferably local) ring $R$ such that ${\rm dim}R>0$ and $R$ has the property that for each $P=Ann_R(r)\in {\rm Min}(R)$ we have $Ann_R …

It's a fact proven by Pendleton, Gilmer, and Ohm (as an obvious corollary of their work, anyways) that PIDs are QR-domains, meaning every overring (ring between the domain and the …

Hi! I'm trying to make headway on a question for my undergraduate honors thesis, specifically the question of which rings of integer-valued polynomials if any satisfy the QR-prope …

I scoured what I could in the literature but I have yet to find the information that should be out there. Consider the property (P1) Every local subalgebra can be embedded in a l …

How to get a Galois extension of the commutative semi-local ring $R=\mathbb{F}_2+v\mathbb{F}_2$, where $v^2=v$. The Galois extension of $R$ of degree $d$ is $\mathbb{F}_a+v\mathbb …

Modular Arithmetic (MA) has the same axioms as first order Peano Arithmetic (PA) except $\forall x (Sx \ne 0)$ is replaced with $\exists x(Sx = 0)$. (http://en.wikipedia.org/wiki/ …

Let $R$ be a commutative ring (with $1$) such that every non-zero divisor in $R$ is a unit (see http://mathoverflow.net/questions/42647/rings-in-which-every-non-unit-is-a-zero-divi …