Search Results

The following are just some partial thoughts about the new question which are too long for comments. Throughout, assume that $R_{\mathfrak{m}}$ is a UFD, $R_0$ is a field, etc... First, let $x\in \ma …

(Note: The original question had an incorrect premise. The ideal generated by elements of positive degree in an arbitrary graded ring is not generally maximal.) Here is the solution to the original p …

Fix $I_2=\langle x^2\rangle \vartriangleleft \mathbb Z[x]$. Let $f=x$. For each polynomial $g\in \mathbb{Z}[x]$ we have $h(1-gf)\in 1+I$ when taking $h=1+gf$. Your question: "Which polynomials $g\in …

Sketch: First, if $e$ is an idempotent in $A$, show that $B$ can be replaced with $B+Re$, and the hypotheses still holds. Use this to reduce to the case that $A$ and $B$ contain the same idempotents …

According to the first paragraph in Shafarevich's paper "On Luroth's problem" (found here http://www.math.ens.fr/~benoist/refs/Shafarevich.pdf) the field of rational functions on the surface $z^2+y^2= …

For this type of problem, I find that free algebras modulo the relations you want often suffice. That is the case here. Take $R=\mathbb{Z}\langle r,s : sr=rs,r^2s=rs\rangle$. We check that $$ r^2(s …

This is just an extended comment: If $Ra$ is essential in a direct summand of $R,$ then $a$ has this property. To see this, fix some idempotent $e\in R$ such that $Ra$ is essential in $Re.$ Then if …

My general rule of thumb is to ask myself the following questions: (1) If I pretend that my paper was written by someone else, and I hadn't seen it before, and I want to verify its accuracy, would …

Several nice characterizations for these rings are worked out as Exercise 4.15 in Lam's book "Exercises in Classical Ring Theory." These include (for commutative rings): (A) $R$ is reduced and $K$-d …

Let us say that an ideal $I\leq R$ has property $(\ast)$, by way of definition, if whenever an intersection of ideals is contained in $I$ then one of the ideals in the intersection is contained in $I$ …

It is a well-known and open problem to determine whether there exists a rectangular cuboid where the distance from any corner to any other corner is an integer. Such a beast, if it exists, is called …

Let $R$ be the subring of $\prod_{i=1}^{\infty}\mathbb{Q}$ of sequences which are eventually constant. This ring has the "obvious" maximal ideals $M_i$ of sequences which are zero in the $i$th coordi …

Suppose $R,S,T$ are rings. Let $_RM_S$ and $_SN_T$ be bimodules. The tensor product $M\otimes_S N$ is naturally an $R$-$T$-bimodule. The "middle" $S$-structure is gone. In your case, $E$ is an $R$ …

In January, Vesselin Dimitrov posted to the arXiv a preprint showing that Mochizuki's work, if correct, would be effective. While this doesn't validate Mochizuki's work it does do a few things: It …

The answer to this quite beautiful question is that there does exist a commutative ring $R$ with $R\cong R[X,Y]$ but $R\not\cong R[X]$. Let $F$ be a field, and take $$ R=F[x_i,y_i,r_i\ (i\geq 0)] $$ …