Centers of Noetherian Algebras and K-theory

I’ll start off a little vauge: Let E be a noncommutative ring which is finitely generated over its noetherian center Z. Denote by modE the category of finitely generated left E-modules and similarly for modZ. We have a functor F:modZ→modE which takes M to E⊗ZM, hence an induced map on (Quillen) K1-groups K1(F):K1(modZ)→K1(modE). I’m interested … Read more

An order in $\mathbb Q[G]$ which is a maximal $\mathbb Z_p$-order in $\mathbb Q_p[G]$ for finitely many primes $p$

Let $G$ be a finite group and $S$ a finite set of prime numbers. I know that every separable $\mathbb Q$-algebra $A$ contains a maximal $\mathbb Z$-order but I wonder if the following is true. Is there a $\mathbb Z$-order $\Lambda$ in $\mathbb Q[G]$ which contains $\mathbb Z[G]$ and satisfies the following two conditions 1) … Read more

When a ring is a polynomial ring?

In the paper (2.11) the authors show that if k∗ is a separable algebraic extension of k and x1,x2,…,xn are indeterminates over k∗ and a normal one dimensional ring A with k⊂A⊂k∗[x1,x2,…,xn] then A has the form k′[t] where k′ is the algebraic closer of k in A. The above is a very strict sufficient … Read more

Is a central simple algebra necessarily cyclic if it splits after a cyclic Galois extension?

Let A be a central simple algebra of degree n over k, dimkA=n2, let K/k be a cyclic galois extension of degree n. Suppose A×kK≅Mn(K), does this imply that A is a cyclic algebra? So the question is in the definition, A is cyclic if it splits after some cyclic extension K contained in A, … Read more

Algebraic-closures of division rings

In what follows, x is always taken to commute with the coefficient ring. This means that for any given polynomial, you can put the coefficients to the right or the left of x as you please. This doesn’t make a difference to the ring itself, but it will make a difference for the roots: a … Read more

Is x∈A1x \in A_1 left algebraic over the subalgebra generated by pp and qq, [q,p]=1[q,p]=1?

Let A1:=A1(x,y,k) be the first Weyl algebra over a field k of characteristic zero, namely, the k-algebra generated by x and y with relation yx−xy=1. Let f:(x,y)↦(p,q) be a k-algebra homomorphism of A1, so [q,p]=1. Denote the image of A1 under f by T (T=A1(p,q,k) is the k-subalgebra of A1 generated by p and q). … Read more

Division in the universal enveloping algebra

Let $\mathfrak g$ be a (semisimple) Lie algebra, $\mathfrak b\subset \mathfrak g$ a Borel and $\mathfrak n = [\mathfrak b,\mathfrak b]$. Then I am interested in solving certain division problems in $U(\mathfrak n)$ on a computer. More specifically, we recall $U(\mathfrak n)$ admits a weight decomposition into components $U(\mathfrak n)[\xi]$. Then given a $P\in U(\mathfrak … Read more

Color algebras and color involutions

If A is a G-graded algebra then one can define on it a color involution, i.e. a bijective linear map preserving the grading such that the image of a product of two homogeneous elements is defined through a bicharacter of the group G. More precisely, ∗:A→A is a color involution if a∗∗=a, for all a∈A; … Read more

Non-commutative rings where every non-unit is contained in a completely prime ideal

Below, all rings are associative and unital; and the word “ideal” always refers to a two-sided ideal. Let’s stipulate that a ring R has property (P) if every non-unit of R is contained in a completely prime(1) ideal. It is well known that, under the axioms of ZFC (say), every commutative ring has property (P):In … Read more

A non-commutative, left duo ring whose only unit is the identity

Let $R$ be a ring (here, rings are always associative, unital, and non-zero). We say that $R$ is a left duo ring if $aR \subseteq Ra$ for every $a \in R$. Question. Is there a non-commutative, left duo ring whose only unit is the identity? It is perhaps worth noting that, if the only unit … Read more