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I think the answer to your question is Section 29 of Matsumura's book "Commutative ring theory". A summary: A complete $p$-ring is a complete local ring whose maximal ideal is generated by $p$. For a …

Here is what I remember from a proof I came up with long time ago (it appeared in some competitions). I am sure it is known, but since the proof is short, I will put it here: The statement can be re …

The old formula for the Frobenius number of a numerical semigroup generated by two elements can be stated as follows: assume $\gcd\{a+1,b+1\}=1$, then the Frobenius number of $S= \left<a+1,b+1\right>$ …

Below are a couple of idle questions that came up one day when I became curious about "matrix factorizations over $\mathbb Z$". Let's start with size $2$: consider the equation $n= ab-cd$ (*), where $ …

This is more like a long comment on Mikhail's answer, which I beg to differ slightly from: Over a regular surface (all local rings are regular, so you don't need to work over fields) and if your divis …

Let $k$ be a field. I am interested in sufficient criteria for $f \in k[x,y]$ to be irreducible. An example is Theorem A of this paper (Brindza and Pintér, On the irreducibility of some polynomials in …

Your question reminds me of the following quote from my advisor, which I can't resist posting here: "Finally I want to remark that the treatment of big Cohen-Macaulay modules here serves as a remind …

Let $a_1,\cdots, a_n$ be integers such that $a_i\geq 2$ for all $i$ and $k>0$ another integer. I am interested in whether there exist integers $x_1,\cdots, x_n$ with $0<x_i<a_i$ satisfying: $$ \fra …

Since, $X^3+Y^3+Z^3-3XYZ=\frac{1}{2}(X+Y+Z)((X-Y)^2+(Y-Z)^2+(Z-X)^2)$, taking $X,Y,Z$ close to each other give some non-trivial and cheap solutions. For instance $(k+1,k,k)$ for $N=3k$, $(k+1,k+1,k)$ …

Let $f(x)=x^3-5x/4$. Then for $x\neq y$, $f(x)=f(y)$ iff $x^2+xy+y^2=5/4$ or $(2x+y)^2+3y^2=5$. The last equation clealy have real solutions. But if there are rational solutions, then there are intege …

Let $[n]$ denote the set $\{0,1,...,n\}$. A subset $S\subseteq [n]$ is said to have complete double if $S+S=[2n]$. Let $m(n)$ be the smallest size of a subset of $[n]$ with complete double. My questio …