**3**

votes

**1**answer

172 views

### Example of proof using the generic matrix

There is a really nice proof of the Cayley-Hamilton Theorem using the generic matrix. I expose it briefly.
One defines the generic matrix $G:=(X_{ij})_{ij} ...

**0**

votes

**0**answers

54 views

### Criterion for global dimension of subring

All rings are assumed to be associative and unital.
If $B$ is a commutative sub-ring of $A$ (which itself needs not be commutative) then what properties of $B$ are both necessary and sufficient for ...

**11**

votes

**2**answers

277 views

### Is the Amitsur-Levitzki identity essentially unique?

Let us consider the matrix algebra. $Mat_n(\mathbb{C})$. The Amitsur-Levitzki identity states that for any matrices $X_1, X_2, ..., X_{2n} \in Mat_n(\mathbb{C})$ the sum $\Sigma_{\sigma \in S_{2n}} ...

**5**

votes

**2**answers

291 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

**0**answers

31 views

### Analogue of Baer's injectivity criterion for comodule algebras

Let $H$ be a Hopf algebra and $A$ an $H$-comodule algebra; denote by $M^H_A$ the category of right $(H,A)$-Hopf modules [i.e. $A$-module, $H$-comodules, everything is compatible with everything else]. ...

**3**

votes

**0**answers

45 views

### dual composition of binary relations

I'm not sure if this is of any interest at all, but I spent some time looking at it a couple of years ago so I'd like to ask for input on this.
Given two binary relations $\rho,\,\sigma$ on a set ...

**4**

votes

**1**answer

321 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 ...

**4**

votes

**3**answers

375 views

### A classification of non-associative algebras with a norm?

I would like to know if there exists any kind of classification of finite dimensional (possibly non-associative) algebras $A$ over $\mathbb R$ that satisfy the following condition:
There exists a ...

**9**

votes

**4**answers

828 views

### What is about J. v. Neumann's “Continuous geometry”?

I am curious about von Neumann's "Continuous geometry", but found no recent text or survey on it. Does anyone know the book and would be so nice to share their impression, and if/how the concept of ...

**20**

votes

**3**answers

839 views

### Two (other) rings…are they isomorphic?

Consider the local rings
$$R = \mathbb{C}[[x,y,z,w]]/\langle xyz+xyw+xzw+yzw\rangle$$
and
$$S = \mathbb{C}[[x,y,z,w]]/\langle xyz+xyw+xzw+yzw+xyzw\rangle.$$
Is $R$ isomorphic to $S$?
Some ...

**1**

vote

**1**answer

62 views

### Near-ring localizations

Are there any known results on localization for near-rings (i.e., "rings" with non-abelian addition and only one-sided distributive law)? The books on near-rings I checked don't mention this topic at ...

**2**

votes

**1**answer

101 views

### Projecting solutions of Hermitian forms over local rings

Let $R$ be a local ring (commutative and with $1$) with maximal ideal $M$, with an involution $\theta$. Let $h$ be a Hermitian form on $R^n$, i.e. $h:R^n\times R^n\rightarrow R$ such that $h$ is ...

**16**

votes

**1**answer

893 views

### Two rings…are they isomorphic?

Edit: I have reverted my question to its original version (which Bjorn Pooenen answered correctly) as requested in the comments.
Consider the local rings
$$R = \mathbb{C}[[x,y,z]]/\langle ...

**1**

vote

**0**answers

39 views

### Bass' stable range for Bezout rings

As discussed in this MO topic, every principal ideal domain has stable rank at most 2. The proof in the accepted answer uses the fact that PID is a unique factorization domain, but there can be no ...

**2**

votes

**1**answer

172 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 ...

**9**

votes

**2**answers

257 views

### A back and forth Euclidean algorithm over the integers--does it have bounded length?

cLet $a,b,c,d\in \mathbb{Z}$ and suppose we have the equation $ac+bd=1$. One way of thinking about this equation is it expresses the fact $\gcd(c,d)=1$. It is well-known that all other similar ...

**2**

votes

**0**answers

110 views

### Is the equational theory of commutative vN regular rings decidable?

I wanted to check whether $A(x,y):=\frac{xy}{x+y}$ is an associative operation in every commutative vN regular ring. Now $A(-1,A(1,1))=A(-1,\frac{1}{2})=1\neq 0 =A(0,1)=A(A(-1,1),1)$. On the other ...

**6**

votes

**2**answers

224 views

### properties of formal delta functions

The formal delta function is
$\,\,\displaystyle\delta(x):=\sum_{n\in\mathbb Z}x^n.
$
If we agree that expressions $(x+y)^n$ for $n\in\mathbb Z$ are always expanded in non-negative powers of the second ...

**7**

votes

**1**answer

141 views

### Can Schwartz-Zippel be formulated for commutative rings instead of fields?

The polynomials which occur in the Schwartz-Zippel lemma could be defined for any commutative ring, yet the lemma is restricted to fields. This makes it inapplicable for ...

**-1**

votes

**0**answers

60 views

### Complement for Jacobson radical [migrated]

Suppose $R$ is a ring with identity and $R$ is of prime power order. Also suppose jacobson radical of $R$ is not trivial. Under which condition there exist a subring $S$ such that
$$R=J(R)\oplus S$$.
...

**2**

votes

**1**answer

194 views

### Graphs of tensoring modules

Let $A$ be a ring, and $L,M,N$ an $A^\mathrm{op} \otimes A$-modules.
$L \otimes_A M$ is then an $A \otimes A^\mathrm{op}$ module so we can tensor it again with $N$ to get an abelian group $(L ...

**13**

votes

**3**answers

964 views

### Continuum Hypothesis

I am new here, so forgive me if this question does not satisfy the protocols of the site.
I know there are so many equivalents to the AC (axiom of choice) and there are books that lists this ...

**10**

votes

**3**answers

420 views

### Natural associative law for a ternary “group”?

Suppose one were to define a group-like structure based on a set $G$
with a ternary (rather than binary) operator $g( a, b, c ) = \left< a, b, c \right>$.
One possible definition for the ...

**0**

votes

**0**answers

74 views

### Is every non-invertible element of a commutative von Neumann regular ring a zero-divisor? (answered: yes!)

As I learned from a previous old question, every commutative von Neumann regular ring is a subdirect product of a family of fields. For a direct product of fields, it seems clear to me that every ...

**14**

votes

**2**answers

498 views

### Nice Algebraic Statements Independent from ZF + V=L (constructibility)

Background and Motivation
I've always been fascinated about algebraic statements independent from ZFC set theory. One such fascinating example comes from considering ...

**1**

vote

**1**answer

169 views

### Self-similarity for simple algebraic structures [closed]

I'm doing this thread because I have some ideas about how to define self-similarity in algebra, but I don't know if this is known at all. Any critics, comments and references are more than welcomed. ...

**5**

votes

**0**answers

150 views

### Operator connected with Hermite polynomials

For $n \geq 1$, define the following operator $M_n$ on the ring of all polynomials with real coefficients.
$$M_n P(x) = nP(x)^2 - x \int_0^x (P'(t))^2 \, \mathrm{d}t$$
Monomials $x^k$ are mapped to $n ...

**5**

votes

**2**answers

163 views

### Can this way of comparing numbers of the form a+b sqrt(K) be generalized?

So I want to make a system for computing with various classes of numbers. One of those is a class of number closed under the standard arithmetic operators ($+$, $-$, $*$ and $/$) along with square ...

**2**

votes

**0**answers

86 views

### Orders of Clifford algebra

Let $C_n$ be the Clifford algebra over $\mathbb{Q}$ associated to negative definite quadratic form $-I_n$ (i.e. $-x_1^2-\dots-x_n^2$). Let $\mathcal{O}$ be a $\mathbb{Z}$-order of $C_n$.
Q1) Is it ...

**1**

vote

**1**answer

115 views

### non unital composition algebra

It is well-known the Hurwitz theorem showing that every unital composition algebra is of dimension 1, 2, 4 or 8 over the base field.
Is there an analogue in the non unital case? ¿Do non unital ...

**9**

votes

**1**answer

233 views

### How do I check that this is a Frobenius algebra?

Let $f_1,f_2,\ldots,f_n\in \mathbb C[z_1,\ldots, z_n]$ be such that the quotient ring
$$A:=\mathbb C[z_1,\ldots, z_n]/(f_1,f_2,\ldots,f_n)$$
is finite dimensional (in other words, it's a ...

**14**

votes

**3**answers

536 views

### Characterize the category of rings

(Sub)categories of many well-studied mathematical objects have been characterized purely in terms of their morphisms. Some (famous) examples:
Sets and functions, due to Lawvere.
Modules over some ...

**3**

votes

**3**answers

183 views

### Modules for an idempotented algebra

Recall that an associative algebra $A$ is called idempotented provided that is the filtered union of subalgebras $eAe$ for $e \in A$ idempotent. I think sometimes people say that $A$ has approximate ...

**0**

votes

**1**answer

51 views

### Reference request for stably free modules

I am looking for some references on the theory of stably free modules. I will call (F) the following property for a ring $R$: every f.g. stably free module over $R$ is free.
1) Is there a standard ...

**7**

votes

**2**answers

209 views

### Two infinite dimensional algebras such that the center of their tensor product is bigger than the tensor product of their centers

I am searching for two infinite dimensional algebras such that the center of their tensor product is bigger than the tensor product of their centers. Who knows of such examples? Thanks a lot.

**0**

votes

**1**answer

151 views

### Discriminant of a polynomial in two variables

I want to compute the discriminant of the following polynomial
$$
F(X,Y)=X^mY^n+\sum_{i=0}^{m-1}\sum_{j=0}^{n-1}c_{ij}X^iY^j.
$$
Here the discriminate means the equation $D(c_{i,j})$ in the variables ...

**8**

votes

**2**answers

756 views

### Spheres over rational numbers and other fields

Let K be an ordered field. Define the n-sphere:
$$S^n(K) := \{ (x_1,x_2,\dots,x_n+1) \in K^{n+1} \mid \sum_{i=1}^{n+1} x_i^2 = 1 \}$$
A set of vectors $v_1, v_2, \dots, v_r \in S^n(K)$ is ...

**38**

votes

**7**answers

4k views

### “Algebraic” topologies like the Zariski topology?

The fact that a commutative ring has a natural topological space associated with it is still a really interesting coincidence. The entire subject of Algebraic geometry is based on this simple fact.
...

**6**

votes

**1**answer

136 views

### Is the following module over a group ring necessarily infinitely generated?

Suppose $\Gamma$ is a (finitely presented, but this is probably irrelevant) group, and $M$ is a finitely generated (EDIT: finitely presented) module over $\mathbb{Q}\Gamma$ which is ...

**2**

votes

**1**answer

93 views

### A quadratic algebra with four generators and four relations

Algebra that I'm going to describe pop-up in my research, it looks completely elementary, but I don't know any appropriate references.
Let $k$ be an algebraically closed field of characteristic ...

**0**

votes

**0**answers

55 views

### The space of sequences of rationals and its dimension [duplicate]

In the following page, I give an example of a vector space not isomorphic to its double dual.
I use the space $E$ of sequences of reals. Its dimension (over the field of the reals) is the one of the ...

**1**

vote

**3**answers

299 views

### smooth connected affine scheme over Z has good reduction almost everywhere

Let $f(x_1,\ldots,x_n)\in\mathbf{Z}[x_1,\ldots,x_n]$ be a polynomial. Assume that the variety cut out by $f$ is smooth and connected (so irreducible) over $\overline{\mathbf{Q}}$. Where can I find a ...

**0**

votes

**1**answer

82 views

### $I$ is principal if and only if $ID_M$ is principal for all $M \in Max(D)$

In a semmi-quasi local domain $D$ ( i.e. $D$ has finitely many maximal ideals),
an ideal $I$ is principal if and only if $ID_M$ is principal for all $M \in Max(D)$.
[See Comm. Rings by Kaplansky, ...

**9**

votes

**0**answers

102 views

### Detecting invertible elements in group rings by their images for finite quotients of the group

Let $G$ be a "nice" infinite group: at least finitely presented and residually finite, maybe also linear and right-orderable (or even bi-orderable, or residually free nilpotent).
Consider an element ...

**15**

votes

**1**answer

362 views

### If $B\subseteq A$ are free & finite rank $R$-algebras, is $R\to A \otimes_B R$ injective?

(In this question, all rings and algebras are commutative with identity.)
I have a situation that boils down to the following data: a ring $R$, an $R$-algebra $A$ with a subalgebra $B$ such that $A$ ...

**11**

votes

**1**answer

201 views

### Hlawka inequality for determinants of positive definite matrices

It is mentioned here that if $A, B, C\in M_{n}(\mathbb C)$ are positive semidefinite, then $$\det (A+B+C)+\det C\ge \det (A+C)+\det (B+C)$$ (quoted from this article) and the special case ($C=\bf 0$) ...

**3**

votes

**2**answers

118 views

### Finite Dimensional Simple nonunital associative Algebras

I have the following problem:
Let K be any field. An finite dimensional associative non-unital algebra A is a vector space A, togeter with a K-biliniear associative operation such that there is NO ...

**5**

votes

**0**answers

199 views

### A generalization of real characters on a group

Yesterday I understood that I can't live without this construction:
Let $G$ be a group, $A$ an associative algebra over $\mathbb R$ and $n\in{\mathbb N}$. We consider a sequence of maps ...

**42**

votes

**1**answer

2k views

### A condition that implies commutativity

Let $R$ be a ring. A notable theorem of N. Jacobson states that if the identity $x^{n}=x$ holds for every $x \in R$ and a fixed $n \geq 2$ then $R$ is a commutative ring.
The proof of the result for ...

**58**

votes

**9**answers

4k views

### Can we unify addition and multiplication into one binary operation? To what extent can we find universal binary operations?

The question is the extent to which we can unify addition
and multiplication, realizing them as terms in a single
underlying binary operation. I have a number of questions.
Is there a binary ...