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      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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      3
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      0answers
      39 views

      Finiteness questions for enveloping algebras

      Let $\mathfrak{g}$ be a Lie algebra over a field of characteristic $0$. The interesting case for these questions is when $\mathfrak{g}$ is infinite dimensional. Is $U(\mathfrak{g})$ a coherent ...
      6
      votes
      0answers
      120 views

      Newer versions of Mahler's Lemma

      I'm trying to find a way to numerically ensure that two constructible numbers are equal (this would be done by a computer). The idea is to find a polynomial $p(x)$ that contains both numbers as roots ...
      4
      votes
      2answers
      182 views

      Complete reducibility and field extension

      Let $\pi$ be a representation of a Lie algebra $L$ in a finite-dimensional linear space $V$ over the field $F$. Let $K$ be a field extension of $F$. Let $\pi_K=\pi\otimes K$ be the corresponding ...
      3
      votes
      1answer
      121 views

      A local ring with a unique minimal ideal

      Let $R$ be a commutative ring with 1 such that it is local with a unique non-idempotent minimal ideal, that is, there are two ideals $I$ and $m$ of $R$ such that for each ideal $K$ of $R$ with $0\not=...
      4
      votes
      0answers
      80 views

      Row rank and column rank of matrix with entries in a commutative ring

      Let $R$ be a unital commutative ring and $A\in M_{n\times m}(R)$. Under which appropriate invariant "rank" of modules discussed in "Ranks of Modules" one can say that the row rank of $A$ is ...
      0
      votes
      1answer
      127 views

      If two monic polynomials of $\mathbb{Z}_p[X]$ (p-adic integer coefficient) are relatively prime modulo p, then do they generate $\mathbb{Z}_p[X]$?

      I'm currently reading a paper of Rene Schoof, and I got stuck in a line. And I'm trying to check the above sentence. Although that seems to be elementary, I hope someone can give me a counterexample ...
      4
      votes
      1answer
      171 views

      List of Casimir elements of low dimensional Lie algebras

      I am interested in explicit formulae for the Casimir elements (or "Casimir operators") of low-dimensional, real, non-Abelian Lie algebras (d=2,3, and possibly 4). I am wondering if there is any ...
      2
      votes
      1answer
      117 views

      Completeness of the ring of Witt vectors

      Suppose that $S$ is a perfect ring of char $p$ and is t-adically complete for some non-zero-divisor $t$. Is the ring of Witt vectors $W(S)$ $[t]$-adically complete? If so can i get a proof please?
      2
      votes
      1answer
      275 views

      Motivation to study the order theory (ring theory)

      I'm currently reading a paper of Georges Gras on the Reflection Principle. The paper uses some theorems about orders (ring theory) from the book "Maximal Orders" by Reiner. I find the book interesting,...
      0
      votes
      1answer
      141 views

      Localization and containment in commutative ring

      Let $R$ be a commutative ring with identity and $x, y $ be fixed elements of $R$ such that for each maximal ideal $m$ of $R$ we have $\langle \frac{x}{1_m}\rangle\subseteq\langle \frac{y}{1_m}\rangle$ ...
      1
      vote
      0answers
      100 views

      Question about Local Henselian Rings

      I have a question regarding properties/characterizations of local Henselian rings exploited in M. Artin's article "On Isolated Rational Singularities of Surfaces": Here the relevant excerpt: Remark: ...
      2
      votes
      0answers
      57 views

      Isoartinian and isosimple modules

      I'm reading this article by A. Facchini and Z. Nazemian, in wich they discuss modules with chain conditions up to isomorphism. A couple of the main concepts are the following: Definition We say that ...
      1
      vote
      1answer
      126 views

      A property for primitive idempotents

      Let $R$ be a (commutative) ring (with identity). A nonzero idempotent $e\in R$ is called primitive idempotent, whenever it has no decomposition into $a+b$ where $a$ and $b$ are nonzero orthogonal ($...
      3
      votes
      0answers
      70 views

      Is a specific endomorphism of $A_1$ an automorphism?

      Let $k$ be a field of characteristic zero, and let $A_1(k)$ be the first Weyl algebra, namely, the associative non-commutative $k$-algebra generated by $x$ and $y$ subject to the relation $yx-xy=1$. ...
      1
      vote
      1answer
      134 views

      Special idempotents in a commutative ring

      Let $R$ be a commutative ring with $1$ and $e$ be an idempotent element of $R$ with the property that if $e=x+y$ (where $x, y\in R$), then there exists $r\in R$ such that either $e=rx$ or $e=ry$. ...

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