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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 ...
        3
        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 ...
        5
        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
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        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 ...
        5
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        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
        98 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 ...
        1
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        0answers
        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
        43 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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