# Tagged Questions

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### The Jordan Plane and Enveloping Algebras

Let $k$ denote a field of characteristic $0$ (assume algebraically closed for convenience). Define $J=k\langle x,y|[x,y]=y^{2}\rangle$. This noncommutative algebra (which can be viewed as a derivation ...
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### Turning left modules into right modules over a homotopy Gerstenhaber algebra

For simplicity's sake, let $A$ be a dg-algebra over $\mathbb{Z}/2\mathbb{Z}$. In the case when $A$ is a commutative algebra, we can turn a left $A$ module into a right $A$ module trivially. Of course ...
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### What do you do if you believe a problem is undecidable?

While the title of this question is subjective, I hope to make what I'm looking for quite concrete. The first, and main question is this: If you believe that a problem you are working on is formally ...
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### How to make the Capelli's identity less mysterious?

The formulation of the Capelli's identity is very elementary; it has important applications in invariant theory and representation theory, see http://en.wikipedia.org/wiki/Capelli%27s_identity To ...
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### Projective dimension of ring over its center

If $A$ is a ring and $Z(A)$ is its center then what is a sufficient condition for the projective dimension of $A$ over $Z(A)$ (ie: $pd_{Z(A)}(A)$) to be finite? (Assuming that $A\neq Z(A)$).
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### Existence of small projective dimensioned modules

Suppose $A$ is a (if necessary unital) associative ring and $I$ is a left ideal in $A$. Let $\operatorname{pd}(M)$ denote the projective dimension of a left $A$-module $M$. Then do either of the ...
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### Non-commutative normalization

Let $A$ be a (non-commutative) associative algebra with 1. Assume that $A$ contains a cental subalgebra $Z$ such that a) $Z$ is a noetherian domain b) $A$ is a finitely generated module over $Z$. ...
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### Finitely generated projective = finitely presented flat over a noncommutative Noetherian ring

Let $R$ be a possibly noncommutative left Noetherian ring and $M$ an $R$-module. I am looking for a reference or a proof for the following fact: $M$ is finitely generated and projective if and only if ...
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### What are the main open problems in the theory of quasigroups and loops?

What are the main open problems in the theory of quasigroups and loops? A short survey would be welcome. Thanks
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### Global dimension of a subalgebra with all units

(All rings here are always assumed to be unital and associative). Setup Let $R$ be a ring, and $A$ and $B$ be $R$-algebras, with $A$ a commutative subalgebra of $B$ satisfying: If $u$ is a unit ...
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### Weyl algebras $A_n(k)$ as tensor product of the first Weyl algebra

In afew threads I've read that the Weyl algebra $A_{n+1}(k)$ is isomorphic to the $k$-tensor product of $A_n(k)$ with $A_1(k)$, why is this true?
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### Poincaré Duality of a quasi-free algebra

I'm completely stumped on this one (yet I feel it is obviously true or obviously false) If $A$ is a quasi-free algebra, then must it satisfy Poincaré duality? All i need to find is a protective ...
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### Explicit calculation of module of derivations on noncommutative polynomial ring

Let $R$ be a commutative unital associative ring and set $R<x,y>$ to be the $R$-algebra of non-commuting polynomials in two variables over $R$. Explicitly how would one go about computing ...
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### Why is “naive” definition of non-commutative spectrum bad?

It is well-known that the category of affine schemes is equivalent to the opposit category of commutative unital rings. So naively, one would think that the same should hold in non-commutative setting....
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### Discriminants of Clifford algebras

I have a Clifford algebra defined over a field of characteristic not equal to $2$. Is there a formula for its discriminant in terms of the corresponding symmetric bilinear form (or in terms of its ...