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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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        242 views

        Functors on the category of abelian groups which satisfy $F(G\times H) \cong F(G)\otimes_{\mathbb{Z}} F(H)$

        Edit: According to the comment of Todd Trimble, I revise the question. What are some examples of functors $F$ on the category of Abelian groups or category of rings which satisfy $$F(G\times H)\cong ...
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        votes
        0answers
        101 views

        Homomorphisms from $k[x,y]$ to $k[x,x^{-1},y]$

        Let $k$ be a field of characteristic zero and let $R_{-1}:=k[x,x^{-1},y]$ be the $k$-algebra of polynomials in $x,y$ containing the inverse of $x$, denoted by $x^{-1}$. Let $f: k[x,y] \to R_{-1}$ be ...
        2
        votes
        0answers
        28 views

        Denominator identity for Lie superalgebras

        Let $\mathfrak g$ be a basic classic simple Lie superalgebra. Fix a maximal isotropic subset $S \subset \Delta$ and choose a set of simple roots $\Pi$ containing $S$. Let $R$ be the Weyl ...
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        vote
        0answers
        78 views

        Could we assume without loss of generality that all coefficients are positive?

        Let $\alpha$ be an element in the group algebra $\mathbb CG$ of a torsion-free group $G$. Assume that, as an operator acting on $\ell^2(G)$, $\alpha$ is positive. Does there exist $\beta\in\mathbb CG$ ...
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        votes
        2answers
        395 views

        Zero tensor product over a complex algebra?

        Let $A$ be an algebra over $\mathbb{C}$. Let $M$ be a left $A$-module, let $N$ be a right $A$-module and consider the tensor product $N \otimes_A M$, which is a complex vector space. Q1: Can this ...
        2
        votes
        1answer
        90 views

        Name and properties of this combination of group algebra and semidirect product?

        Given a field $k$, a group $G$, and a homomorphism $\phi : G \to \mathrm {Aut} (k)$, we can define a ring $\widehat {k [G]}_\phi$ as follows: As an abelian group it is isomorphic to the group algebra $...
        2
        votes
        1answer
        95 views

        Field of definition of a finite dimensional division algebra and how to reduce it

        Let F be a field, and E/F an infinite algebraic extension. Let D be a finite dimensional division algebra over E (meaning its center is also E). Is it possible to somehow gow down to a finite ...
        2
        votes
        1answer
        239 views

        When does the forgetful functor from modules to vector spaces have a right adjoint?

        Given any algebra $R,$ when does the forgetful functor $R\text{-}Mod \rightarrow Vec$ have a right adjoint? Does this imply any finiteness conditions on R? Is there a book/paper discussing this? I'...
        3
        votes
        1answer
        77 views

        A “concrete” example of a one-sided Hopf algebra

        I came to know from the paper Left Hopf Algebras by Green, Nichols and Taft that one may consider a Hopf algebra whose antipode satisfies only the left (resp. right) antipode condition. To be more ...
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        votes
        0answers
        100 views

        Contractible Banach algebras

        A Banach algebra $A$ is contractible if $H^1(A, X)=0$ for all Banach $A$-bimodules $X$. Now to my question Let $A$ be Banach algebra and $I$ be closed ideal of $A$. If $I$ and $A/I$ are both ...
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        votes
        1answer
        236 views

        Example of a ring where every module of finite projective dimension is free?

        I'm interested in seeing an example of a ring which is not self-injective where every module admitting a finite projective resolution is free, or at least projective. Note that self-injectivity says ...
        3
        votes
        1answer
        122 views

        Multiplication in $Z(\mathbb{C}S_n)$ [duplicate]

        I am trying to multiply two generators of center $Z(\mathbb{C}[S_n])$ of ring algebra of symmetric group of $n$ elements. We know that these generators are given by sums of conjugacy classes in $S_n,$ ...
        2
        votes
        0answers
        42 views

        Generalizing polynomial identities for rings

        For a ring $R$, a polynomial identity of $R$ is a polynomial (in non-commuting variables) $f(x_1,\ldots,x_n)\in \mathbb{Z}[x_1,\ldots, x_n]$ such that for any choice of $a_i\in R$, $f(a_1,\ldots, a_n)=...
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        votes
        0answers
        154 views

        Homotopy quotient of groups

        Suppose $0\to A \stackrel{\iota}{\to} B \stackrel{\pi}{\to} C \to 0$ is a short exact sequence of groups. We have an induced map $k[\iota] : k[A] \to k[B]$ of group algebras over a field $k$. What ...
        2
        votes
        0answers
        77 views

        Open sets on a Stone space

        If $B$ is a Boolean algebra (possibly assumed complete), is there a standard name for the Heyting algebra (or frame) $L := \Omega(S(B))$ of open sets on the Stone space $S(B)$ of $B$, — or for the ...

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