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

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2answers
188 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 ...
2
votes
3answers
136 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 $x * y = 0$ ...
6
votes
1answer
99 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 ...
1
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0answers
45 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 ...
5
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0answers
336 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
votes
1answer
201 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 ...
11
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7answers
731 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 ...
4
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0answers
121 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
65 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 ...
15
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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 ...
2
votes
1answer
194 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 ...
4
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1answer
184 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
78 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 ...
7
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0answers
218 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 ...
6
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0answers
144 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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0answers
86 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?
3
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0answers
112 views

Isomorphic bound quiver algebras for different admissible Ideals

We know that for a path algebra KQ, whether or not KQ is finite dimensional (namely, Q may or may not have oriented cycles), there might be different admissible ideals I and J of KQ for which the ...
4
votes
1answer
366 views

Re-interpreting vector spaces in a choice-less model of ZF as modules over a regular ring in ZFC

I am searching a module $M$ over a (von Neumann) regular ring $A$ ($\forall a\in A$, $\exists x\in A$: $axa=a$) with two properties: (1) every finitely generated submodule of $M$ is projective ...
1
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1answer
197 views

Existence of a left adjoint to tensor product implies projectivity

Let $S$ and $R$ be two (not necessarily commutative) $k$-algebras for $k$ a field. If I have a $S$-$R$ bimodule $_SM_R$, I can form the functor $_SM_R\otimes_R (-):R\text{Mod} \rightarrow ...
0
votes
1answer
81 views

Norm of a number in an algebraically closed field

I consider an algebraically closed field of characteristic zero $F$ as a vector space over a real closed field $R \subset F$. I would like to define a norm on $F$ in an invariant fashion, i.e. if we ...
5
votes
1answer
248 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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1answer
152 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 ...
2
votes
1answer
219 views

Trace of finitely generated projective module

Let $k$ be a field and $A$ a $k$-algebra with unit. The trace module is $$ T(A)=A/[A,A], $$ where $[A,A]$ is the left $A$-module generated by all elements of the form $ab-ba$ for $a,b\in A$. The ...
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0answers
77 views

Simplest (?) example of bicrossed product Hopf algebra

Suppose we have two Hopf algebras, H and A and additionally A is (left) H-module algebra and H is (right) A comodule coalgebra. This means that A is left module over H and moreover that ...
3
votes
2answers
314 views

Are algebraic structures uniquely identifed by their free objects?

It might be a naive question, as I am not a specialist in this field. This is a follow-up to this question. I want to study varieties of objects generalizing ordered monoids, in particular using an ...
3
votes
2answers
434 views

Examples of complete distributive lattices that are not Heyting algebras

Here is a short question with a possibly simple and short answer: I need an example of a complete distributive lattice that is not a Heyting algebra which should be an infinite complete lattice that ...
0
votes
0answers
68 views

approximate coordinates in a one dimensional lattice

suppose I have a finite set of real numbers ${r_1, \ldots r_n \in \mathbf{R} }$ and a single real number $x \in \mathbf{R}$. Is there a fast algorithm for finding integer numbers ${i_1, \ldots i_n \in ...
4
votes
1answer
159 views

Potentially identity elements in an Abelian group

I didn't see this problem before. I motivated by the questions Is every commutative group structure underlying at least one (unitary, commutative) ring structure A basic question about rings ...
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0answers
113 views

Non commutative analogy of compact-open topology

Let $R$ be a ring, define a topology on $AUT(R)$(Or End(R)) with the following subbase: For every two 2-sided Ideal $I$ and $J$, a subbase element is $B(I,J)=\{f\in AUT(R) \mid f(I)+J=R\}$. We can ...
1
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1answer
244 views

A question in ring theory

Is there an example of two groups $G_{1}, G_{2}$ such that there are two non isomorphic ring $R_{1}$ and $R_{2}$ such that the additive group of both rings is isomorphic to $G_{1}$ and their unit ...
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2answers
310 views

A basic question about rings

Perhaps this is a trivial question, but I have no idea how to justify it. Call a pair of groups $(G_1, G_2)$ ring-compatible if $G_1$ is abelian and there exists a ring $R$ with addition and ...
3
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2answers
176 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 ...
1
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0answers
116 views

subring of the matrix algebra

Let $Mat_2(\mathbb{Z})$ be the $\mathbb{Z}$-algebra of $2\times2$ matrices with integer entries. Let $A$ be a $\mathbb{Z}$-submodule of $Mat_2(\mathbb{Z})$ containing $\mathbb{Z}$. We want to show ...
3
votes
1answer
205 views

On rings $R$ for which $R \cong \frac RI$ for any proper two-sided ideal $I$

This is a problem I asked in SE, but it seems the question is more suitable for MO. Consider a ring $R$ (not necessary with identity or commutative) such that for any proper two-sided ideal $I$, ...
2
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0answers
56 views

Cyclic modules over serial rings

Let $R$ be a serial ring. What can be said about the uniform dimension of cyclic $R$-modules? Specially I would like to know Is it true that every cyclic $R$-module has finite uniform dimension ?
4
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0answers
99 views

Sum of projective submodules of a projective over a semihereditary ring

Sorry in advance if this is too silly. Let $R$ be a right semihereditary ring and $P$ a projective right $R$-module. It is well-known that finitely generated (thus projective) submodules of $P$ form a ...
2
votes
2answers
201 views

How much information does the multiplicative semigroup of an algebra contain?

How much do we know about an given algebra when we only know its semigroup strucure under the product law? How far can two algebras be distinguished by knowing only their semigroup strucure? The ...
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0answers
132 views

How to find $n$ such that the group of units $U(\mathbb{Z}/n\mathbb{Z})$ has a given abelian subgroup?

Given an integer $n$, we can determine the structure of the multiplicative group of integers modulo $n$ ($U(\mathbb{Z}/n\mathbb{Z})$) by the factorization of $n$. Hence we can easily find all the ...
5
votes
1answer
223 views

Inverse limit of Gorenstein local rings is again Gorenstein?

If we have the system of surjective ring homomorphisms $f_{i,i+1}: R_{i+1} \twoheadrightarrow R_i$ for an arbitrary $i \geq 0$ such that all $R_i$ are Gorenstein local ring. Let us put ...
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0answers
87 views

Super-Gorenstein ideal of ${\Bbb F}_p[[X_1,\ldots,X_n]]$

Let $A \colon= {\Bbb F}_p[[X_1,\ldots,X_n]]$ be a $n$-variable power series ring over a finite field ${\Bbb F}_p$. We put ${\frak m}_A \colon= (X_1,\ldots,X_n)$. Definition(Super-Gorenstein ideal): ...
7
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2answers
340 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 ...
0
votes
2answers
289 views

Dual of a module

Let $M$ be a $ \mathbb{Z}_{p}[[T]] $-module and $X=Hom(M,\mathbb{Q}_{p}/\mathbb{Z}_{p})$ be the dual of $M$. Let $X[p^n]$ denotes the $p^n$-torsion points of $X$. Is $X/X[p^n]$ the dual of $M[p^n]$ ...
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2answers
455 views

Does base extension reflect the property of being isomorphic?

Let L/K be a (separable?) field extension, let A be a finite dimensional algebra over K, and let M and N be two A-modules. Let $A' = L \otimes_K A$ be the algebra given by extension of scalars, and ...
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0answers
65 views

What is a necessary and sufficient condition that the kernel of a semi-module homomorphism is a partitioning sub-semi-module?

I would like to identify a representation of the subcategory of a comma category of semi-rings, whose objects are abelian group objects. When attempting to identify the representation, the following ...
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0answers
64 views

An invariant number of modules over Auslander Gorenstein modules

Given an Auslander Gorenstein $R$ of injective dimension $\mu$, one can associate with each finitely generated module $M$ with a number $\varepsilon_{\mu}(M)$, which is the number of the ...
1
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1answer
98 views

When for every module $M$, $|E(M)| = |M|$

Is there a non-semisimple ring $R$ such that for any left $R$-module $M$, $|E(M)| = |M|$ ? (where $E(M)$ is the injective hull of $M$ and $|M|$ is the cardinality of $M$)
1
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1answer
159 views

“order two sequence” in a paper of Waldhausen

In F. Waldhausen's paper "Algebraic K-theory of generalized free products, Part I", page 142,line 19, there is a term "order two sequence". Can anyone explain its meaning to me? According to the ...
5
votes
2answers
260 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 ...
0
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2answers
260 views

Rank of a $ \mathbb{Z}_{p}[[T]] $ module

Let $p$ be a prime and $M$ is a finitely generated $ \mathbb{Z}_{p}[[T]] $ module. Suppose $M[p]$ denotes the $p$-torsion of $M$. Then $M[p]$ and $M/(p)$ are both $ F_{p}$ vector spaces. So we can ...
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0answers
101 views

Universally catenary and all its formal fibers over minimal members are Cohen-Macaulay but it has a nonCohen-Macaulay formal fiber

Please help me to find a Noetherian local ring $R$ such that: $R$ is universally catenary and all its formal fibers over minimal members of $Spec(R)$ are Cohen-Macaulay but $R$ has a nonCohen-Macaulay ...