Recall that Kneser conjecture (now Lovasz theorem) claims that if the family of $k$-subsets (subsets of cardinality $k$) of given $(2k+d)$-set $M$, $d\geq 1$ are colored into $d+1$ …

First, pick a commutative ring $k$ as the "ground field". Everything I say will be $k$-linear, e.g. "algebra" means "unital associative algebra over $k$". Then recall the followi …

Can anyone suggest a relatively gentle linear algebra text that integrates vector spaces and matrix algebra right from the start? I've found in the past that students react in ver …

Let $f: R \to S$ be a morphism of Noetherian rings (or more generally $S$ can just be an $R-R$ bimodule with a bimodule morphism $R \to S$). Suppose $f$ is faithfully flat on both …

D is a central division algebra over F. We know that we can always find a maximal subfield K inside D such that K/F is separable. I want to know can we always make it Galois?

I (Anton) have edited this question to be the question Pete and Zeb discuss in the first few comments. What conditions on a ring $R$ imply that the units of $R[x]$ are exactly …

Of course, commutative algebra is a fundamental topic in algebraic geometry, number theory, representation theory, and so on. However, there are some instances (most obviously tro …

I found myself "naturally" dealing with an object of this form: X is a complex vector space, with a "product" (a,b) → {aba} which is quadratic in the first variable, linear i …

Let $R$ be a (noncommutative) ring. (For me, the words "ring" and "algebra" are isomorphic, and all rings are associative with unit, and usually noncommutative.) Then I think I k …

All the statements of the Archimedean property with which I am acquainted fundamentally uses ℕ -- more than as a totally ordered semi-group, really being the 'standard model' …

Can you offer some examples of such rings, other than $\frac{k[x,y]}{(x^{2}, xy)}$. Thanks.

This questions is inspired by an exercise in Hungerford that I have only partially solved. The exercise reads: "A domain is a UFD if and only if every nonzero prime ideal contains …

Let's say I have my favorite finite dimensional algebra $A$, and favorite module $T$. Now assume that the reason $T$ is my favorite module is that it has a cool property: there …

In a vector space $V$ over a field $F$, a nonempty subset $W$ of $V$ is a subspace if it is closed under addition and scalar multiplication. For a module $M$ over a ring $R$ with i …

Suppose we define an arbitrary field extension $K/F$ to be Galois if, for all subextensions $L$ of $K/F$, we have $K^{\operatorname{Aut}(K/L)} = L$. In words: for any element $x$ o …