Non-commutative rings and algebras, non-associative algebras, universal algebra and lattice theory, linear algebra, semigroups.

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Extensions on Higher-dimensional local fields

I have the following question: Let $M/L$ b a finite extension of n-dimensional local fields and $t_1,\dots, t_n$ a system of local parameters of $L$ with valuation $v$. Let us fix an $1\leq i \leq ...
1
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0answers
27 views

Valuations in Higher-dimensional local fields

I have the following question which I believ should be true but I would like to have a different opinion about it: Let $M/L$ is a finite Galois extension of $n$-dimensional local fields and ...
0
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0answers
39 views

Reductive Group Actions and Completion

Suppose that $A$ is a Noetherian (not necessarily commutative) $\mathbb{C}[h]$-algebra equipped with a rational action of an affine reductive group $G$, i.e., $A$ is a $\mathbb{C}[G]$-comodule and ...
7
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1answer
159 views

Why can't one modify Kaplansky's proof to conclude that every projective module is a direct sum of its finitely generated projetive submodules?

Due to the examples given in the answer to this question, I know that the conclusion is of course incorrect. But by reading Kaplansky's proof of theorem 1 in this paper and replacing every occurrence ...
1
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1answer
156 views

symmetric algebra of an ideal and syzygies

Let $(R,\mathfrak{m})$ be a Cohen Macaulay local ring and $I=(a+1,\ldots,a_g,a_{g+1})$ an almost complete intersection ideal of codimension $g.$ Let $R^k\longrightarrow R^{g+1}\longrightarrow ...
3
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1answer
195 views

Metrics and completions on the direct limit of matrices of all sizes over arbitrary fields

Let $M_n$ denote the $n$ by $n$ matrices. Consider the homomorphisms $$\phi_{n,kn}: M_n \rightarrow M_{kn}$$ which takes a matrix $A \in M_n$ to $A \otimes I_k \in M_{kn}.$ This gives a sensible way ...
4
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1answer
287 views

Topological Problems Solved by Lattice Duality

It is well known the success of lattice dualities (as Pontryagin duality for abelian groups, Stone duality for Boolean algebras and Priestley duality for distributive lattices) to solve algebraic ...
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0answers
81 views

Possibilities for dimensions of $\mathfrak{m}^i/\mathfrak{m}^{i+1}$ for a local ring

Let $R$ be a local commutative ring with maximal ideal $\mathfrak{m}$, and denote by $k$ the residue field $R/\mathfrak{m}$. Then we can look at the sequence of $k$-vectorspaces $$R/\mathfrak{m}, ...
1
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1answer
100 views

Universal constructions that factor through endomorphisms

If $\cal A$ is a variety of algebras (e.g., all groups) and $\cal B$ is a subvariety defined by some set of identities $X$ (e.g., abelian groups with $X = \{xy \simeq yx\}$), then there is a functor ...
64
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11answers
3k views

Can a non-surjective polynomial map from an infinite field to itself miss only finitely many points?

Is there an infinite field $k$ together with a polynomial $f \in k[x]$ such that the associated map $f \colon k \to k$ is not surjective but misses only finitely many elements in $k$ (i.e. only ...
1
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1answer
78 views

Weyl algebras $A_n(k)$ as tensor product of the first Weyl algebra

In afew threads I've read that the Weyl algebra $A_{n+1}(k)$ is isomorphic to the $k$-tensor product of $A_n(k)$ with $A_1(k)$, why is this true?
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28 views

Grassmann algebra morphism with universal property

I'm pretty sure that the following doesn't work, but nevertheless i wanted to ask, maybe this is a kind of well-known construction i've never heard of: Let $\Lambda(\mathbb{R}^n)$ be a finite ...
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1answer
43 views

How to find a matrix by its characteristic value and characteristic vectors? [closed]

Now I am studying linear algebra course, In that for a given matrix we are finding the characteristic values (eigen vlaues) and characteristic vectors (eigen vectors). But my qustion is why cant we ...
1
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1answer
134 views

for which truth-operations f can f-membership in a prime ideal be represented by a polynomial?

Nota bene: all rings are supposed commutative with $1$ and all ring homomorphism should be unital. Let $B$ be the set of truth values {True, False}. For a formula $\phi$ denote by $[\phi]\in B$ it's ...
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2answers
129 views

Is there an intuitionistic generalized boolean algebra (of Stone)?

A "boolean algebra without the greatest element" was called by Stone "generalized boolean algebra" and he axiomatized it. Is there any publication about "preudo-boolean algebras without the greatest ...
1
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1answer
58 views

Does a BCL algebra define a partial order?

A BCL algebra is a universal algebra with a binary operation denoted as "$*$" and a $0$-ary operation (constant) denoted as "$0$", satisying the following axioms: (1) $x * x = 0$; (2) if $x * y = 0$ ...
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2answers
336 views

Is the class of inverse semigroups globally determined?

This question is a follow-up to this one I asked on math.stackexchange. I've decided to ask here because I believe this is a research-level question. I'm sorry if I'm wrong -- I'm not a researcher ...
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0answers
42 views

Alternative Generating Sets of the Quantum Special Linear Group

The Hopf algebra ${\cal O}_q(SL(N))$ has as generators the elements $u^i_j$, subject to certain $q$-relations. Moreover, the antipode is bijective, with square equal to a multiple of the identity on ...
6
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1answer
68 views

Infinite dimensional simple algebras of finite degree

Let $F$ be a field and let $B$ be an $F$-algebra. The degree of $B$ over $F$ is the smallest positive integer $\deg_F B = d \geq 1$ such that every element of $B$ satisfies a (monic) polynomial of ...
21
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5answers
694 views

How to recognize a Hopf algebra?

Suppose we are handed an algebra $A$ over a field $k$. What should we look at if we want to determine whether $A$ can or cannot be equipped with structure maps to make it a Hopf algebra? I guess in ...
1
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0answers
41 views

On generating Euler Square of index q, q-1 (where q is any prime power)

Can somebody help me in generating Euler Square of index q, q-1 (where q is any prime power). Also, kidly tell me if there is any code availeble for generating the stated Euler Square. The details of ...
3
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1answer
180 views

Proving that a semigroup is regular

In a number of diverse situations of interest to me (mostly associated with something called the abelian sandpile model), one can define a nonabelian semigroup generated by commuting elements ...
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0answers
103 views

Cycles in Quivers and Path Algebras

I cannot find anything giving the algebra of a quiver with a single cycle on three or more vertices. In other words if your quiver consists of n vertices (n>2), and e_i is connected to e_{i+1} (taking ...
4
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0answers
245 views

The logarithm over $\mathbb F_1$

In 'Cyclotomy and analytic geometry over F1', Manin proposes a version of the notion of `analytic function' over the 'field with one element $\mathbb F_1$'. Question 1: can somebody explain or give ...
5
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1answer
219 views

A generalized K- theory via generalized idempotents

Edit After the answer by Neil Strickland, I add the word "a ring" in this new version. In the literature, there is a concept of generalized idempotent: an n- idempotent is an element $a$ of a Banach ...
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6answers
528 views

Properties of rings that have an elegant description in terms of the associated category of modules

Suppose $A$ is a ring. Then $A$ happens to be a division ring iff every left $A$-module is free. (See here for proofs). I think this is very beautiful; what other properties of rings have an elegant ...
5
votes
1answer
163 views

Equivariant Formality

Let $G$ be a finite group and $\mathcal{A}$ be a $dg$-algebra. Assume $G$ acts on $\mathcal{A}$, i.e. there exists a homomorphism $G\to {\rm Aut}_{dg}(\mathcal{A})$. Assume further there exists a ...
5
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2answers
183 views

Integration on Compact Semirings

I want to know if integration of functions $f:X\rightarrow G$ where $G$ is a compact semiring has been defined and if it is possible to ensure that all continuous functions are integrable. This is ...
12
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7answers
3k views

What are the reasons for considering rings without identity?

I think a major reason is because Lie algebras don't have an identity, but I'm not really sure.
3
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0answers
93 views

Formal DG-algebras

Sorry for this question but I really have difficulties with model categories. Usually a $dg$-algebra $A$ is called formal, if there exists a $dg$-algebra $B$ and quasi-isomorphisms $$A\leftarrow B\to ...
3
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0answers
144 views

Comparing different Euclidean algorithms on a Euclidean domain

I have posted this question at stackexchange (502413), without responses until now. In the papers by T. Motzkin: The Euclidean Algorithm, Bull. AMS 55, 1949, pp. 1142--1146 and P. Samuel: About ...
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5answers
1k views

An algebra of “integrals”

When discussing divergent integrals with people, I got curious about the following: Is there an $\mathbb{R}$-algebra $A$ together with a map (could be defined on just a subspace) $$\int_0^{\infty}: ...
3
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1answer
144 views

Properly “transfinitely” Euclidean domains

Are there integral domains which admit ordinal-valued Euclidean functions but not $\mathbb{N}$-valued Euclidean functions?
3
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4answers
607 views

When does a planar ternary ring uniquely coordinitise a projective plane?

From every projective plane a coordinitisation can be constructed on a planar ternary ring, and conversely from every planar ternary ring a projective plane can be constructed. (For background see ...
6
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3answers
422 views

IBN for algebraic theories

Let us say that a finitary algebraic theory $\tau$ has IBN (invariant basis number) if the free functor $F : \mathsf{Set} \to \mathsf{Mod}(\tau)$ reflects the isomorphism relation: If $S,T$ are sets ...
3
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0answers
62 views

Quasi-isomorphisms and Subalgebras

Let $A$ and $B$ $dg$-algebras over $\mathbb{C}$. If there exists an isomorphism $f:A\to B$, then every subalgebra $A'$ of $A$ is isomorphic to the subalgebra $f(A')$ of $B$. What is if $f$ is ...
3
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2answers
153 views

Examples of cancellative normal semigroups

I've got a couple of things to test against normality in cancellative semigroups. A normal semigroup $S$ is one in which for any $x\in S$ we have $xS=Sx.$ This implies the Ore condition $$x,y\in ...
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2answers
237 views

Non-commutative reduced rings of order $p^2$

Let $p$ be a prime number. Is there a non-commutative reduced ring of order $p^2$? (Note that any ring of order $p^2$ with identity is commutative).
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5answers
2k views

Origin of exact sequences

I have seen exact sequences appearing a lot in algebraic texts with different purposes. But I've never seen names of the people associated with it. Also I don't understand what's so good about showing ...
18
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3answers
936 views

Freeness of a Z[x]-module

Definition: Call a mapping $f: \mathbb{Z} \rightarrow \mathbb{Z}$ a generalized polynomial if for any distinct integers $m$ and $n$ we have $(m - n)|(f(m)-f(n))$. It is easy to check that polynomial ...
2
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1answer
125 views

Jacobson radical and group rings/subalgebras

Let $G$ be a finite group and $N\le G$ be a subgroup. Consider the group algebra $kN$ as a subalgebra of $kG$ over an algebraically closed field $k$ of positive characteristic. What can we deduce ...
2
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1answer
133 views

coherent modules

Let $R$ be a nontrivial ring. A right $R$-module $M$ is called coherent if ${\rm Ker} (f)$ is finitely generated for any $R$-module homomorphism $f: L\to M$ with $L$ finitely generated. It is ...
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0answers
66 views

Units in residue classes

Let $K$ be a CM-field of degree $2n$. (Quadratic extension of totally real number field) Let $\mathcal{O}_K$ be the ring of integers in $K$, and $m\geq 1$ an integer. Let $U_K$ be the group of units ...
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16answers
18k views

Linear Algebra Texts?

Can anyone suggest a relatively gentle linear algebra text that integrates vector spaces and matrix algebra right from the start? I've found in the past that students react in very negative ways to ...
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0answers
127 views

What happens to simple modules under Ringel duality?

If $A$ is a quasi-hereditary algebra then its Ringel dual $A'=End_A(T)$ is the endomorphism algebra of a (minimal) full tilting module $T$ for $A$. The algebra $A'$ is again quasi-hereditary, although ...
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2answers
665 views

Finite local rings

There is some classification of finite commutative local rings. For example how many not isomorphic finite local rings with the same order $p^k$ and the same residue field $\mathbb F_p$ exist?
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7answers
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What makes a theorem *a* “nullstellensatz.”

I know what the (Hilbert) Nullstellensatz says. A MathSciNet search on "nullstellensatz" turns up nearly 200 papers, with only a minority offering either new proofs or new applications of the classic ...
4
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0answers
108 views

Which endomorphism algebras are not Morita-trivial?

Let $\mathcal C$ be a symmetric monoidal category — for example, $\mathcal C$ might be the category of $R$-modules for a commutative ring $R$. At a minimum, I request further that: $\mathcal C$ is ...
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2answers
345 views

“Inverse problem” for Brauer groups

This question is just a curiosity, but I'm really interested in the answer. It was originally posted on math.stackexchange ...
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0answers
69 views

Relationships between finiteness of stable rank and IBN property of rings

Does any ring of finite stable rank have IBN property? Where can we find this result?