# Questions tagged [noncommutative-rings]

Questions about rings that are not necessarily commutative.

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### Unital subrings of simple Artinian rings

Let $S=M_n(D)$, the ring of $n\times n$ matrices with entries in a division ring $D$. Now suppose that $R$ is a simple unital Artinian subring of $S$. Is it the case that $R\cong M_k(D')$ for some ...
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### Submodule of matrix space is free

Let $\mathcal{H}$ be an infinite dimensional complex inner product space, and denote by $\mathcal{H}^{n \times n} = \mathcal{H} \otimes_{\mathbb{C}} \mathbb{C}^{n \times n}$ the corresponding matrix ...
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### Noncommutative algebra question inspired by Young-Jucys-Murphy elements

This is really a followup to Why are Jucys-Murphy elements' eigenvalues whole numbers? , specifically to Igor Makhlin's beautiful answer. I'm trying to make it even more beautiful by getting rid ...
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### Nullstellensatz for maximal left ideals of quantum plane

Let $R=\mathbb{C}\langle x,y\rangle/\langle xy=qyx\rangle$ be the quantum plane algebra. Does some sort of Nullstellensatz holds for the maximal left ideals of $R$? By this we mean all maximal left ...
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### Primitive elements in the universal enveloping algebra of Lie superalgebra

Let $\mathfrak{g}$ be a Lie superalgebra over $\mathbb{C}$. Denote by $U(\mathfrak{g})$ the universal enveloping algebra of $\mathfrak{g}$. We know that there is a natural super Hopf algebra structure ...
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### Abelian groups such that $A \cong \mathrm{End}(A)$ and "complete rings"

Motivation: for any ring $R$ there is the natural monomorphism $\mathrm{in} \colon R \to \mathrm{End}(R_{add}): r \mapsto (x \mapsto rx)$, where $R_{add}$ is an additive abelian group ( rings are ...
1 vote
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### Pseudo-coherent complexes over sheaves of non-commutative rings

I am posing a question on derived categories to which I was not able to find an answer anywhere in the literature. I would appreciate any answer, hint or suggestion. Assume that $\mathcal{R}_X$ is a ...
48 views

Let $A$ be a $k$-algebra for some commutative ring $k$. Recall that $A$ is said to be separable over $k$ if the multiplication map $A\otimes_k A^{\operatorname{op}}\to A$ has a section as a map of $A\... 4 votes 0 answers 107 views ### A non-commutative, left duo ring whose only unit is the identity Let$R$be a ring (here, rings are always associative, unital, and non-zero). We say that$R$is a left duo ring if$aR \subseteq Ra$for every$a \in R$. Question. Is there a non-commutative, left ... 5 votes 1 answer 212 views ### Rings s.t. each element has a power lying in the center (and their completely prime ideals) Let$R$be a ring (throughout, all rings are associative and unital). We say$R$satisfies condition (C) if, for every$a \in R$, there exists an integer$n \ge 1$(depending on$a$) such that$a^n$... 9 votes 1 answer 613 views ### Existence of a finite extension of ℤ providing a finite extension of the primes Let$R$be a ring (possibly noncommutative with zero-divisors). A non-unit and non-zero-divisor element$r \in R$will be called irreducible if for all$a,b \in R$such that$r=ab$, then$a$or$b$is ... 4 votes 1 answer 219 views ### Is a non-degenerate finite-dimensional algebra unital? Let$A$be a finite-dimensional (not necessarily unital) associative algebra over the field of complex numbers$\mathbb{C}$(but I'm also interested in more general fields). Assume the multiplication ... 2 votes 2 answers 153 views ### Module complements to rings embedded in a projective module Let$R$be noncommutative unital ring and$M$a projective (right)$M$-module. Assume that$R$embedds into$M$as a right -module. A) If$R$is a semisimple ring, then of course$R$admits an$R$-... 1 vote 1 answer 120 views ### Nilpotent elements of index$2$in group algebra$FA_4$Let$A_4 = K_4 \rtimes C_3$be alternating group on$4$symbols and$F$be finite field containing$4$elements. By definitions of group algebra and augmentation ideal, there exist a natural map $$\... 4 votes 1 answer 187 views ### Infinite linearly independent set in finitely generated module Let R be a (commutative, otherwise the answer is easy, see the comment below) ring and let M be a finitely generated R-module. Is it possible that M admits an infinite linearly independent set?... 2 votes 0 answers 55 views ### Decomposition of augmentation ideal in a group ring Let R be a ring and G be a finite group with invertible order in R. Assume that the augmentation ideal \Delta (G) of group ring RG has a decomposition M_1\oplus M_2\oplus \cdots M_k as an ... 2 votes 1 answer 84 views ### Every module of finite uniform dimension is a direct sum of (finitely many) indecomposable Crossposted on StackExchange on July 28 (no answer so far). Let R be a (commutative or non-commutative, associative, unital) ring. It is well known that any artinian or noetherian R-module M can ... 26 votes 3 answers 655 views ### Subtraction-free identities that hold for rings but not for semirings? Here is a concrete, if seemingly unmotivated, aspect of the question I am interested in: Question 1. Let a and b be two elements of a (noncommutative) semiring R such that 1+a^3 and 1+b^3 ... 1 vote 0 answers 51 views ### Class groups and zeta functions for maximal orders in CSAs I'm looking for references for certain algebraic objects in the context of maximal orders in (finite dimensional) central simple algebras over algebraic number fields. Does anyone know of any good ... 2 votes 0 answers 81 views ### Do r(a) \leq^\oplus R and r(a) = r(a^2) imply r(a) = eR and aR \subseteq (1-e)R for some idempotent e? Let R be a (commutative or non-commutative, associative) ring with unity, and let a be an element of R such that r(a) = r(a^2), where r(\cdot) denotes a right annihilator. It follows that r(... 1 vote 0 answers 34 views ### Rings where every indecomposable principal right ideal is extensive Let R be a (commutative or non-commutative, associative) unital ring. Following Nicholson & Yousif [1, p. 21], we say that a right ideal \mathfrak i of R is extensive if every R-linear ... 2 votes 2 answers 91 views ### Every non-zero submodule of R_R has an indecomposable direct summand: True when R is von Neumann regular? Let's say that a (right) module M is well complemented if every non-zero submodule of M has an indecomposable direct summand (by the way, is there a better or more standard name for this property?)... 1 vote 0 answers 101 views ### On the rings R with the property that eR \cong fR for all primitive idempotents e, f \in R Let R be a (commutative or non-commutative) ring with identity. As usual, an idempotent e \in R is primitive if eR (the principal right ideal generated by e) is indecomposable as a right ... 1 vote 1 answer 138 views ### Origins of a theorem on an atomic factorizations in domains and cancellative monoids satisfying the ACCPL and the ACCPR Let H be a (commutative or non-commutative) monoid. We say that H satisfies the ACCPL (ascending chain condition on principal left ideals) if there exists no infinite sequence of principal left ... 14 votes 1 answer 772 views ### Is there any non-commutative ring such that every element other than the identity is a zero divisor? A (unital) ring R with the property that every element other than the identity 1_R is a (two-sided) zero divisor, seems to be commonly called a "0-ring" or "\mathcal O-ring"... 6 votes 1 answer 184 views ### Is there any structural characterization of the rings in which every element other than the identity is a (two-sided) zero divisor? [I fear that I'm missing something obvious here, but I'll dare to ask anyway.] As we all know, a division ring is a (unital, associative, non-zero) ring where every non-zero element is a unit. So, let ... 2 votes 2 answers 175 views ### An algebra which is a direct sum of simple sub-bimodules over a subalgebra Let A be an infinite-dimensional noncommutative algebra over a field, let B be an infinite-dimensional subalgebra of A, and let A be a direct sum of projective simple B-sub-bimodules. Then ... 1 vote 1 answer 302 views ### Are there algebras over reals besides complex numbers, where identities, analoguous to (-1)^i=e^{-\pi} and i^i=e^{-\pi/2} hold? Are there algebras over real numbers (with exponentiation), such that there is such z that does not include components in \mathbb{C} (or in a subset isomorphic to \mathbb{C}), for which (-1)^z\... 2 votes 1 answer 86 views ### The ring of upper triangular n-by-n matrices over a skew field is (left and right) Rickart Let T_n(R) be the ring of upper triangular n-by-n matrices with entries in a (commutative or non-commutative) unital ring R. It happened to me to note that, if R is a skew field, then T_2(R)... 3 votes 1 answer 75 views ### Terminology for a ring satisfying the DCC on chains of principal right ideals generated by the powers of an element Question. Is there any standard name for a (commutative or non-commutative) unital ring R with the property that, for every a \in R, the (descending) chain R, aR, a^2 R, \ldots, is eventually ... 5 votes 1 answer 140 views ### Do r(a) \leq^\oplus R and r(a) = r(a^2) imply r(a) = eR and r(1-a) \subseteq (1-e)R for some idempotent e? Let R be a (commutative or non-commutative) unital ring, fix a \in R, and denote by r(\cdot) the right annihilator of an element. Question. If r(a) is a (right) direct summand of R and r(a)... 8 votes 1 answer 346 views ### If the Grothendieck ring of a semiring on a free commutative monoid is unital, is the original semiring unital? Suppose S is an associative semiring whose underling commutative monoid is free (in particular, cancellative) and that its Grothendieck ring G(S) is a unital ring. Can we conclude that S must be ... 1 vote 1 answer 73 views ### If, in a unit-regular ring, the right annihilator of a equals the right annihilator of b, then aR = bR? Recall that a (unital) ring R is von Neumann regular (VNR) if, for each x \in R, there exists y \in R such that x = xyx; and unit-regular if such an element y can be taken to be a unit. ... 9 votes 1 answer 140 views ### Special nilpotent elements Let R be a (noncommutative, associative) ring. Set N_2:=\{x\in R : x^2=0\}, the set of nilpotent elements of degree 2 (also called the square-zero elements). If x,y\in R satisfy xy=0, then ... 5 votes 0 answers 146 views ### Representation theory terminology question For a paper I'm writing, I need a term for a representation-theoretic concept that I'm sure someone has thought of before, so I thought I'd ask here rather than just make something up. Let G be a ... 4 votes 1 answer 243 views ### H^1(X, GL(n, \mathcal{O}_X)) and Vector Bundles Let X be a nice space: a manifold, a variety over a field, or so on. We know that line bundles on X up to isomorphisms are given by H^1(X,\mathcal{O}_X^*). Can it be generalized to higher rankal ... 0 votes 1 answer 231 views ### Given a unitary commutative ring R, what are the rings R\langle x,y\rangle/(x^2-A,y^2-B,yx-a-bx-cy-dxy) called We are studying the rings$$ R \langle x, \, y \rangle\,\big/\left(x^2-A, \, y^2-B, \, yx-a-bx-cy-dxy \right) $$Do you know if they have a name? 1 vote 1 answer 135 views ### Do you know a finite unitary reversible ring that is not isomorphic to its opposite? And the minimal with that property? Do you know a finite unitary reversible ring that is not isomorphic to its opposite? And the minimal with that property? The examples of rings not isomorphic to their opposite that I know of are not ... 4 votes 0 answers 154 views ### Do you know rings without involutions, auto-anti-isomorphics? In that case, what is the minimal example? Do you know rings without involutions, but auto-anti-isomorphic (isomorphic to their opposite)? In that case, what is the minimal example? If a ring has an involution f, then f is an anti-automorphism;... 5 votes 1 answer 69 views ### Reference for a certain derivation on the ring of ordered series over a free monoid Let R be a (commutative or non-commutative) unital ring, X be a non-empty set, and R \langle\! \langle X \rangle\! \rangle be the ordered series ring (in fact, a ring of formal power series over ... 1 vote 2 answers 241 views ### How many non-isomorphic associative algebras of dimension 2 over the field F_{p^k} are there? How many non-isomorphic associative algebras of dimension 2 over the field F_{p^k} are there? As much as I have searched, I have not found any results that answer my question; not even for k = 1,2. 2 votes 1 answer 72 views ### Separability of \mathbb{C}[x,y_1,\ldots,y_r] over \mathbb{C} + (h,y_1,\ldots,y_r) The answer to this MO question says the following: Lemma 1. Let h \in \mathbf C[x] be a polynomial of degree n \geq 2. Then \mathbf C+(h) \subseteq \mathbf C[x] is unramified if and only if h ... 7 votes 2 answers 297 views ### Idempotent Laurent polynomials (in noncommuting variables) Let K be a field and R=K\langle X_1,\dots,X_n,X_1^{-1},\dots,X_n^{-1}\rangle the Laurent polynomial ring in n noncommuting variables. Can R have idempotents distinct from 0 and 1? 1 vote 0 answers 180 views ### Road map: beyond Artin-Wedderburn theorem For a noncommutative semisimple ring R, its structure and its category of representations can be largely understood using Artin-Wedderburn theorem. Such structure theory is useful, for example, in ... 3 votes 0 answers 52 views ### Reference for NIM-rep theory for non-commutative fusion rings? The literature on nonnegative integer matrix representations (NIM-reps) seems to be focused on commutative fusion rings, since a primary motivation there is for rational conformal field theory (RCFT). ... 5 votes 1 answer 463 views ### Do you know which is the minimal local ring that is not isomorphic to its opposite? The most popular examples are non-local rings and minimal has 16 elements. I am interested in knowing examples of local rings not isomorphic to their opposite. 4 votes 1 answer 134 views ### What is the extended centroid of a free algebra? For a prime ring R, you can define its "Martindale ring of quotients" Q(R). See for example: Martindale, Wallace S. III, Prime rings satisfying a generalized polynomial identity, J. ... 6 votes 1 answer 279 views ### Algebra with a certain abelian group as the multiplicative group Let A be an abelian group. Are there an algebra \mathfrak{X}(A) s.t. the multiplication group is isomorphic to A ? i.e.$$ \mathfrak{X}(A)^{\times} \simeq A.$$For example, for$A=\mathbb{Z}/4\...
Let $M$ be a free right $R$-module. When $M_R\cong R_R^n$ with $n\in \mathbb{Z}_{\geq 1}$, then we know that the endomorphism ring $E={\rm End}(M_R)$ is isomorphic to $\mathbb{M}_n(R)$. We also know ...
Let $R, S$ be Noetherian $k$-algebra, where $k$ is a field, and $P \otimes S$ is Noetherian. let $P$ be a prime ideal of $R \otimes S$ such that $P \cap (R \otimes 1) = (0) = P \cap (1 \otimes S)$, ...