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Questions tagged [noncommutative-algebra]

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For questions about rings which are not necessarily commutative and modules over such rings.

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Let $D$ be a division ring with a total order $\le$ which is compatible with addition and multiplication, i.e. for any $a, b, c \in D$ \begin{align} &a \le b &\implies& a + c \le b + c \\ &...
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I say that a skewfield (division ring) $D$ over $\mathbb{Q}$ is cyclotomic whenever $D$ admits a finite $\mathbb{Q}$-basis $\{\zeta_1,\dots,\zeta_n\}$ with each $\zeta_i$ a root of unity (i.e., $\...
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I'm studying commutative ring theory from Hideyuki Matsumura's book, but it's so abstract that I try to come up with lots of concrete examples on my own---for instance, classifying commutative ...
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Is there any good reference (including proofs) in english for a proof of the theorem stated in the title? I'm interested in maximum number of non zeros independent vector fields over spheres and non ...
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Let $F$ be any field, we call a $F$-algebra $(a,b)_F$ for $a,b \in F^{\times}$ a quaternion algebra over $F$ a ring containing $F$ which is a $4$-dimensional $F$-vector space having a basis $1, i, j,k$...
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I noticed that integration by substitution is essentially the ``sprinkler rule'' (distributivity of a composition over an action). I wrote an article where: The integral is treated as a monoid ...
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I'm stuck on the following exercise from a qualifying exam: Let $A$ be an Artinian ring and $M$ an $A$-module. For $f$ an $A$-endomorphism of $M$ satisfying $f(M) \subseteq J(A) M$ (where $J(A)$ is ...
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By a ring I mean an associative ring with multiplicative unity, and by a division ring I mean a skew field. My question is as follows Let $D, S$ be two division rings and $_DM_S$ be a $(D, S)$-...
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Let $H$ be a finite index subgroup of $G$ with coset decomposition $G=\bigsqcup_{j=1}^mx_jH$. Let $A$ be a left $\mathbb{Z}H$-module. Then since $\mathbb{Z}G$ is a $(\mathbb{Z}G,\mathbb{Z}H)$-bimodule,...
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If $R$ is a commutative ring, $p(x)$ is a polynomial over $R$ and $r$ is an element of $R$, then $p(x)$ is irreducible if and only if $p(x + r)$ is irreducible. If I understand it correctly, the proof ...
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Is the following statement true? Let $M$ be a $R$-module, $G \subseteq M$ a generating set, and $L \subseteq M$ a linearly independent set. Then $|L| \le |G|$. I know the result is true in finite-...
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Let $R$ be a finite ring, and let $G$ be a finitely generated group. For simplicity, assume that $ G = \langle x, y \rangle $ is generated by two elements. Consider the group ring $R[G]$, and let $I$ ...
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Let $R$ be a Noetherian ring, and let $M$ be a finitely generated Noetherian torsion $R$-module. Suppose $a \in M$ is a non trivial element. Question: Is it possible that there exist infinitely many ...
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It is known that a group ring $R[G]$ is a unique factorization domain (UFD) if and only if $R$ is a UFD and $G$ is a torsion-free abelian group. Question: Has there been any study into the conditions ...
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Let G be a real Lie group and H be a closed subgroup. The Green's imprimitivity theorem tells us that the crossed product algebras $\mathbb{C} \rtimes H$ and $\mathcal{C}_0(G/H) \rtimes G$ are Morita ...
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Some important classes of commutative rings can be represented as rings of continuous functions of a certain type. For example, by Gelfand duality every commutative unital C*-algebras is isomorphic to ...
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I have read that the procedure of forming field of fractions of a noncommutative ring $R$(...more generaly to adapt the general machinery of localizations to non commutative rings/modules) is much ...
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Let $R$ be a regular local ring and $n>0$ be an integer. Consider the matrix ring $A:=M_n(R)$. I am wondering if $\text{Hom}_R(A,R)$ is a projective $A$-module? By Lemma 2.15 it is projective as a ...
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I was reading this paper by G. Tabuada. Here in section 2.2 author ask to notice every quasi equivalence of dg categories is a Morita equivalence. From the definition it seems that quasi equivalence ...
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To prove that every $SO(3)$-invariant $C^\ast$-algebra structure in $C^\infty(S^2)$ must be commutative [theorem 7.1, Commun. Math. Phys. 122, 531-562], Rieffel gets a contradiction based on an ...
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Let $k$ be a field and consider the free algebra $R:=k\langle x,y \rangle$ over a field $k$. Is this ring left (or right) hereditary (i.e. every left (or right) ideal is a projective $R$-module)?. If $...
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This is from S. Lang, Algebra, 2002. We are given a field $k$ and a finite group $G$ such that $n := \vert G \vert; \ \text{char } k \nmid n$. Then by the Artin-Wedderburn theorem, we have a ...
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Let $R$ be a ring. Let $G$ be the group of invertible matrices with entries in $R$. Is there a way to check if a given matrix $M \in G$ can be expressed as a commutator? i.e. there esists $A, B \in G$,...
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An element $r$ in a ring is quasi-unipotent if it satisfies $(r^k - 1)^n = 0$ for some $k, n$. Let $R$ be a ring. Suppose we have a unit $a \in R^*$ which is not quasi-unipotent. Is it possible that ...
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Let $F(x_1, \dotsc, x_n)$ be the free group over $x_1, \dotsc, x_n$. Define a word $w = x_{i_1}^{k_1} \dotsm x_{i_m}^{k_m}$ in reduced form to have length $m$ and size $\lvert k_1 \rvert + \dotsb + \...
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I'm reading the 1980 paper "Graph Algebras" by KH Kim, L Makar-Limanov, J Neggers, and FW Roush. It is a prerequisite to the solution to the isomorphism problem for graph groups aka right-...
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Let $R$ be a ring (unital, associative, not necessarily commutative) and let $I$ be a two sided ideal. Define $I_0=I$, and now for any ordinal $\alpha$, having defined $I_{\beta}$ for all $\beta<\...
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Let $K$ be a commutative ring and let $R$ be a $K$-algebra (i.e. a unital, associative, not necessarily commutative ring with a morphism $K\to Z(R)$). Then $R$ is said to be a PI-algebra over $K$ if ...
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Suppose $0\longrightarrow M'\longrightarrow M\longrightarrow M''\longrightarrow0$ be an exact sequence. If $M$ is locally noetherian then, $M',M''$ are locally noetherian. I have proved that $M'$ is ...
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Let $K$ be a field with involution $\bar{}$. Let $(V,b)$ be a reflexive space. That means $V$ is a $K$-vector space and $b$ is a symmetric or skew-symmetric non-degenerate sesquilinear form. If $\bar{}...
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Let $(A,\ast)$ be a semisimple algebra with anisotropic involution, and let $\rho:A \to \text{End}_K(V)$ be its $\ast$-representation. Let the decomposition of $V$ into the direct sum of simple left $...
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Let $A_1,\dots,A_n,B_1,\dots,B_m$ be simple rings such that $$A_1\times\cdots\times A_n\cong B_1\times\cdots\times B_m,$$ I wish to conclude that $n=m$ and $A_j\cong B_j$ after a permutation. The ...
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Let $R= \left\{ \left ( \begin{matrix} \mathbb{Z}_{2} & \mathbb{Z}_{2}\\ 0 & \mathbb{Z}_{2}\\ \end{matrix} \right ) \right\}$ be a ring. We know that if $Q$ is its classical right ring of ...
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The following theorem is from Lorenz's "Algebra, Volume II: Fields with Structure, Algebras and Advanced Topics", p. 167, F20, about a characterization of a separable field extension using a ...
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I'm a masters student in algebra. I've seen two versions of results giving specific situations where the (left) global dimension of a ring can be computed as the projective dimension of a specific ...
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A ring $R$ is said to be right McCoy if whenever $fg=0$ for nonzero $f,g\in R[x]$, there exists $r\ne 0$ in $R$ such that $fr=0$. Left McCoy rings defined similarly. A ring that is both left and right ...
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I am teaching myself noncommutative ring theory and have started reading T.Y. Lam's A First Course in Noncommutative Rings, but I find it quite difficult. What are the prerequisites for this book? I ...
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Let $A, B$ be $n$ by $n$ matrices with entries in a field $k$. When is it the case that $t A + s B$ is nilpotent for formal indeterminates $t, s$? This condition is equivalent to $\operatorname{span}...
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Let $R$ be a non-commutative ring with unity, and let $I$ be a two-sided ideal of $R$. If $I = eR$ for some idempotent element $e$ (i.e., $e^2 = e$), can we conclude that $I = Rf$ for some idempotent ...
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Let $k$ be a commutative ring and let $R$ be an associative $k$-flat algebra. I was reading about Hochschild cohomology and it is given actually by $$HH^{\ast}(R)\colon = Ext^{\ast}_{R\otimes_{k} R^{...
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Crossposted on Mathematica SE Is there a systematic/algorithmic/computational way to "complete the square" or find a sum-of-squares decomposition for a trace of non-commuting operators? ...
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Let $\mathbb{Z}_p$-algebra $R$ be an integrally closed domain of characteristic zero. For a finite abelian group $G$, we have $G=H \times P$, where $P$ is the unique Sylow p-subgroup. How could we ...
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I already asked two questions about groupring $\mathbb{F_2}[\mathcal{Q_8}]$ but this question is slightly different. Let $\mathbb{F_2}$ be the ring of integers modulo 2 and $\mathcal{Q_8}$ be the ...
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Let $\mathbb{F_2}$ be the ring of integers modulo 2 and $\mathcal{Q_8}$ be the group of quaternions such that $\mathcal{Q_8}=\lbrace \pm1,\pm i, \pm j, \pm k\rbrace$. Then $\mathbb{F_2}[\mathcal{Q_8}]$...
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Let $R$ be a ring with unity. Symmetric ring: A ring $R$ is symmetric if for any $a,b,c \in R$ satisfy $abc=0$ then $acb=0$. Reversible ring: $R$ is reversible if $ab=0$ implies $ba=0$. ...
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Let $k$ be a field and $V$ be a countable dimension vector space over $k$. Let $R:=\operatorname{End}_{k}(V)$. By https://ringtheory.herokuapp.com/rings/ring/15/, we know R is a von Neumann regular ...
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Consider a compact Hausdorff space space $X$, a $\sigma-$algebra of Borel sets $\mathcal{B}(X)$ and probabilistic Radon measure $p$ on $X$. Let $L^\infty(X,\mathcal{B}(X),p;\mathbb{C})$ be a ...
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I don't understand an identification $$ gr^F(\widehat{A})=gr(A)[t^\pm] $$ from Lemma 1.1.20 in Ginzburg notes https://gauss.math.yale.edu/~il282/Ginzburg_D_mod.pdf and I would like to understand why ...
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I need to prove, that any $R$-module $M$ such that any surjective morphism $f: M \twoheadrightarrow N$ splits is semisimple. I suppose the easiest way to do it is to show that any submodule of $M$ is ...
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Question: Let $G$ be a finite group. Let $W$ be an irrep of $G$ over $\mathbb R$ such that $W^{\mathbb{C}}\cong V \oplus V$ for some complex irrep $V$ of $G$. Is it true that a primitive (minimal) ...
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