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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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      Lie structure over $R$-module

      In Higgins' paper Baer invariants and the Birkhoff-Witt theorem (J. Algebra 11 (1969) 469–482, doi:10.1016/0021-8693(69)90086-6) the following definition is given: A Lie structure over the $R$-module ...
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      votes
      0answers
      52 views

      Free skew fields over sets of different cardinal

      Let $K$ be a field and let $X$ be a set. Denote by $\mathcal D_K(X)$ the free skew $K$-field on $X$. Assume that $|X|\ne |Y|$. Is it true that $\mathcal D_K(X)$ and $\mathcal D_K(Y)$ are not ...
      3
      votes
      1answer
      127 views

      Complete local rings, automorphisms and approximation

      Consider two local morphisms $f,g: B\rightarrow A$ of noetherian complete local rings and $f$ surjective. Does there exist an integer $n\in\mathbb{N}$, such that if $f=g \mod \mathfrak{m}_{A}^{n}$ ...
      3
      votes
      0answers
      55 views

      Simple coalgebra under base change

      Let $C$ be a simple coalgebra over a field of characteristic $0$. Let $K$ be a field extension of $k$. Is the coalgebra $C\otimes_k K$ over $K$ simple?
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      vote
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      31 views

      Coinvariant of algebra

      If $G$ acts on $C_c^{\infty}(G)\otimes C_c^{\infty}(G)$ diagonally by right translation, is the coinvariant $(C_c^{\infty}(G)\otimes C_c^{\infty}(G))_{R(G^{\Delta})}$ isomorphic to $C_c^{\infty}(G)$ ...
      2
      votes
      1answer
      86 views

      A weak Schur's lemma for non-semisimple finite dimensional algebras

      Let $B \subseteq C$ be an inclusion of finite dimensional (associative) algebras over a field $k$. Assume that $C$ is a free $B$-module. Let $\bigoplus_i U_i$ be a decomposition of $B$ into ...
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      votes
      1answer
      154 views

      Definition of a Dirac operator

      So it seems that a Dirac operator acting on spinors on $\psi=\psi(\mathfrak{su}(2),\mathbb{C}^2)$ can be written in this case simply as: $D=\sum_{i,j} E_{ij}\otimes e_{ji}$, where $E_{ij}$ are ...
      2
      votes
      0answers
      114 views

      Noether’s “set theoretic foundations” of algebra. Reference

      In [C Mclarty] we read [Noether] project was to get abstract algebra away from thinking about operations on elements, such as addition or multiplication of elements in groups or rings. Her algebra ...
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      votes
      0answers
      70 views

      Any f.p. faithful simple module over a primitive group ring?

      Recall that a ring $R$ is primitive if it has a faithful simple left module. Let $G$ be a countable discrete group and $R=\mathbb{k}G$, where $\mathbb{k}$ denotes some field or $\mathbb{Z}$. There ...
      3
      votes
      1answer
      99 views

      Infinite dimensional finitely generated algebraic division algebra

      Is there a division algebra $D$ with center $K$ that satisfies the following 3 conditions? 1) $D$ is of infinite dimension over $K$; 2) every element of $D$ is algebraic over $K$; 3) $D$ is ...
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      vote
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      156 views

      Primes of the power series rings

      Let $A_n \colon= K[[X_1,\ldots,X_n]]$ be a $n$-variable formal power series ring. By setting $X_n \mapsto 0$, we obtain a natural surjection \begin{equation*} \psi_{n,n-1} \colon A_n \...
      4
      votes
      0answers
      61 views

      A presentation for a subalgebra

      Let $K$ be a field, and let $I=(g_1,\ldots, g_r)$ be an ideal in $A:=K[X_1,\ldots ,X_n]$. Let $\{f_1,\ldots f_m\}$ be a subset of $A$, and let $B$ be the $K$-subalgebra of $A$ generated by $f_1,\...
      3
      votes
      0answers
      44 views

      Embedding the Mészáros subdivision algebra in an Orlik-Terao localization

      The following is an open question (Question 4.1) from my paper $t$-Unique Reductions for Mészáros's Subdivision Algebra (published version in SIGMA 2018, and slightly updated preprint version with ...
      2
      votes
      2answers
      332 views

      Lehmer’s totient problem

      Euler’s totient function $\varphi$ is a function defined over $\mathbb{N}$ so that $\varphi(n)=|\{m\mid m<n\wedge (m,n)=1\}|$. Now Lehmer’s totient problem asks whether $n$ is prime iff $\varphi(n)...
      3
      votes
      0answers
      103 views

      Injective resolution of the ring of entire functions

      Let $R$ be the ring of entire functions on $\mathbb{C}$. I heard that the concrete value of the global dimension of the ring depends on continuum hypothesis. I would think that the injective dimension ...

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