# Questions tagged [ra.rings-and-algebras]

Non-commutative rings and algebras, non-associative algebras, universal algebra and lattice theory, linear algebra, semigroups. For questions specific to commutative algebra (that is, rings that are assumed both associative and commutative), rather use the tag ac.commutative-algebra.

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### Elementary equivalence for rings

Let $\mathcal{L}$ be a first-order language, and $M$ and $N$ be two $\mathcal{L}$-structures. We say that $M$ and $N$ are elementarily equivalent (write $M \approx N$) if they satisfy the same first-...
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### Some folklore about crystaline rings of differential operators

This question is a follow up to my previous question on rings of crystaline differential operators, to which I refer for the adequate definitions. First, let's consider the case of an algebraically ...
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### An open problem about simple Noetherian rings

The following is a well-known open problem in ring theory (see, for instance, Goodearl, Warfield, 'An introduction to noncommutative Noetherian rings, Appendix, Problem 19) Question: Let $R$ be a left ...
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### Expositions of the classical approach to local class field theory (Brauer group and Hasse invariant)

I've posted this question already on MSE and didn't get much out of it, so I hope it's OK to repost here. I'm an undergraduate trying to learn local class field theory from the corresponding chapter ...
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### Connections in non-commutative geometry

Let $K$ be a field, $A$ a unital associative $K$-algebra and $M$ a left $A$-module. A connection on $M$ is a $K$-linear map $\nabla:M\to \Omega^1A\otimes_AM$ which satisfies the Leibniz rule. ...
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### How can I define and construct a module of integers of Grassmann variables?

This is a question that I asked in mathematics stack exchange : I am interested about constructing a certain space where the coordinates are "integers" of grassmann variables. I know that ...
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### Rings where all indecomposable modules are projective or injective

Let $A$ be a semi-perfect noetherian ring. Is there a nice classification of such $A$ such that every indecomposable finitely generated $A$-module is projective or injective? Im also interested in ...
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### Relation between enveloping algebras and algebras of differential operators

I asked this question on math stack exchange about 3 years ago, but received no answer. Our base field $\mathsf{k}$ will be algebraically closed of zero characteristic. Let $X$ be an smooth affine ...
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### Wedderburn–Artin like theorem for infinite dimensional Lie algebras?

The Wedderburn–Artin Theorem is one of the cornerstones of the structure theory of (associative) rings. Wedderburn–Artin Theorem : Let $R$ be a left Artinian ring with zero Jacobson radical. Then $R$ ...
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### Subalgebras of finite extensions

Suppose that $A\subset B$ is a finite extension of rings. Is it true that every intermediate extension $A\subset C\subset B$ finite over $A$?
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### Eigenvalues of a subset of matrix semigroup

My apologies for slightly longer post but I wanted to explain lower dimensional cases and their proofs before asking the actual question, which starts after the phrase The general case below. A two-...
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### Prove that $\Bbb C[x,y]/(x^3+y^3-1)$ is not a UFD

I am posting this question on MO since I haven't received any answers on MSE. Below is my (very elementary) attempt. Feel free to post a solution using facts in algebraic geometry and facts about ...
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### Link invariants from Hecke relations of higher order

Alexander theorem says oriented links in $\mathbb{R}^3$ can be represented by closures of braids. Markov theorem says that braids related by Markov moves produce isotopic braid closures, and vice ...
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### Strassen factorization for $n \geq 3$ [migrated]

There is the famous Strassen matrix multiplication method (see here for further information). In essence it boils down to the fact, that we can multiply 2x2 matrices with each other and don't have to ...
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### Direct product of groups are isomorphic, but factor groups are not isomorphic [migrated]

Question: $A,B,C$ are groups and we know $A\times B\simeq A\times C$. Is $B$ isomorphic to $C$? My work: (1) If $A,B,C$ are finite Abel groups, then this proposition is true, because we just need to ...
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### Is there a "cohomology theory" for involutive algebras?

I'm aware that there are cohomology theories for algebraic structures related to involutive algebras (or "involution algebras" or "$*$-algebras" if you prefer those terms) like Lie ...
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### The associated graded algebra of a finite dimensional algebra

$\DeclareMathOperator\rad{rad}$Let $A$ be a finite dimensional algebra (we can assume that $A \cong KQ/I$, for a quiver $Q$ and an admissible ideal $I$ if that helps). Denote by $A_G$ the associated ...
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### Reference request for the smallest set( resp. abelian group) that a group ( resp. ring ) has a faithful action on

For a proof of the Cayley's Theorem, it is obvious to see that a group ( resp. ring ) has a faithful action on itself by left-multiplication. I would like to extend the result for a bit and find the ...
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