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

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10
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2answers
342 views

Computing intersection of subrings

Let $R$ be a finitely generated commutative ring over a field, for concreteness. If $S,T \leq R$ are two finitely generated subrings, is their intersection also finitely generated? (Certainly ...
1
vote
0answers
72 views

Normalization (integral closure) of $\mathbb Z_p[x]$ in function field of a curve to obtain Model of curve

I want to follow this construction of a normal model of a curve: Let $p\neq 2,3$ and $Y\to \mathbb P¹$ be a smooth projective curve over $\mathbb Q_p$ with function field $L/\mathbb Q_p(x)$ e.g. ...
3
votes
0answers
101 views

Certain conditions on cancellative semigroups

This is extracted from this question following Benjamin Steinberg's suggestion. For a semigroup $S,$ let $P(S)$ denote the power semigroup of $S,$ which is made up of all non-empty subsets of $S$ ...
6
votes
1answer
256 views

Rings satisfying the polynomial equation $x^4=x^2$

There are many results concerning the commutativity of rings satisfying a polynomial equation. I want to know if there is any result/reference about (finite) rings that satisfy the polynomial equation ...
4
votes
1answer
256 views

Homomorphisms from projective modules

Let $B$ be a $A$-algebra which is free of finite rank as $A$-module. Let $X$ be a finitely generated projective left $B$ module. (So $X$ is also a f.g. projective $A$ module.) Are these homomorphism ...
0
votes
0answers
67 views

An embedding of modules by tensor product over a Noetherian domain

I have a problem on Ring theory. I would like to prove or disprove the following statement: Let $R$ be a Noetherian domain. Then by the Goldie theorem $R$ have $Q$ as a full ring of quotients and $Q$ ...
5
votes
0answers
97 views

Centralizers of elements in free group algebras

Let $A$ be a group algebra of a free group, and $x \in A$. What is the centralizer of $x$? Is there something like Bergman's theorem for free associative algebras?
2
votes
1answer
248 views

Automorphisms of ideals of $\mathbb{C}[t]$

Let $f(t)\in\mathbb{C}[t]$, and let $I_f$ be the ideal in $\mathbb{C}[t]$ generated by $f(t)$. The ideal $I_f$ has a natural $\mathbb{C}$-algebra structure. My question is the following: For ...
7
votes
3answers
389 views

Ring of differential operators of a quotient ring

All rings are assumed to have unity. Let $k$ be a field. Recall the definition of Grothendieck's ring of ($k$-linear) differential operators $D(R;k)$ of a commutative $k$-algebra $R$: ...
4
votes
1answer
226 views

Cancellable elements of a power semigroup

For a semigroup $S,$ its power semigroup $P(S)$ is the semigroup of all non-empty subsets of $S$ with the operation given by $AB=\{ab\,|\,a\in A,b\in B\}.$ I would like to know about the cancellable ...
3
votes
1answer
92 views

Existence of small projective dimensioned modules

Suppose $A$ is a (if necessary unital) associative ring and $I$ is a left ideal in $A$. Let $\operatorname{pd}(M)$ denote the projective dimension of a left $A$-module $M$. Then do either of the ...
3
votes
1answer
184 views

Classical deformation of algebras

Given a complex manifold (or a smooth scheme) $X$, the classical (infinitesimal) deformation theory is parametrized by the first cohomology with coefficients in the tangent sheaf $H^1 (X, T_X)$. ...
3
votes
0answers
186 views

Is the “algebraic closure” of the quaternions, finite dimensional? [closed]

This post is a sequel of: What's the algebraic closure of the quaternions? $\mathbb{H}$ is algebraically closed for the polynomials of the form $\sum a_r x^r$, but it is not for the polynomials ...
8
votes
1answer
256 views

Division algebras over extension fields / reducibility of $G$-modules

Reformulation of the question (see below for the original question): Let $K$ be an algebraic number field and $D$ a finite-dimensional $K$-division algebra. Is there a description of the field ...
3
votes
1answer
97 views

Extension of an involutive automorphism

Suppose that $g$ is a complex semi-simple Lie algebra and $g'$ its reductive subalgebra. If $\tau$ is an involutive automorphism of $g'$, can $\tau$ be extended to an involutive automorphism of $g$ ...
3
votes
1answer
279 views

Strong Morita equivalence and representation theory

In the context of pure algebra we say that two algebras (in general: rings) $A,B$ are Morita equivalence when there are bimodules $_AP_B,_BQ_A$ such that $P \otimes_B Q \cong _AA_A$ and $Q \otimes_A P ...
2
votes
1answer
245 views

Example of a Frobenius algebra that is not projective over a Frobenius subalgebra

I'd like to know an example of a Frobenius algebra $A$, with a subalgebra $B$ that is itself a Frobenius algebra, such that $A$ is not projective as a left $B$-module. I don't require any ...
1
vote
0answers
76 views

(Affine) Schemes and the point of view of morphisms with values in a field

Let $A$ be a commutative ring with unity. If $\varphi_1 : A \rightarrow K_1$ and $\varphi_2 : A \rightarrow K_2$ are ring morphisms from $A$ to fields, $\varphi_1$ and $\varphi_2$ are said to ...
3
votes
3answers
364 views

Polynomials of low degree that clone polynomials of higher degree

Let $f(x_1,\dots,x_{16})=(x_1+x_2+x_3+x_4)(x_5+x_6+x_7+x_8)(x_9+x_{10}+x_{11}+x_{12})(x_{13}+x_{14}+x_{15}+x_{16})\in\Bbb R[x]$. Let $\mathcal{Z}$ be the zero set of $f$ in ...
6
votes
1answer
204 views

Purely noncommutative algebra-Morita equivalence

Morita equivalence of algebras certainly don't preserve commutativity: even if $A$ is commutative there are plenty of noncommutative algebras which are Morita equivalent with $A$---for example all ...
0
votes
0answers
42 views

Comparing least degrees of certain polynomials

Let $\Bbb K$ be an infinite or a finite field with $\mathsf{char\mbox{ }}\Bbb K\neq 2$ and let $M\subsetneq\Bbb K[x_1,\dots,x_n]$ be the set of multilinear polynomials. Fix $S\subsetneq\{0,1\}^n$ and ...
8
votes
3answers
625 views

Does a left basis imply a right basis, without AC?

If $_DV_D$ is a $D$-$D$-bimodule, and we have a $D$-basis for $V_D$, do we still need AC to get a $D$-basis for $_DV$? (The original question appears below. But this shorter question gets at the ...
0
votes
0answers
95 views

Dimensions of two spaces

Let $R=\Bbb K[x_1,\dots,x_n]/I$ and $Z=\mathsf Z(I)$ where $\Bbb K$ is any field with $char(\Bbb K)\neq 2$. Is there a way to describe space of $f_1,f_2\in R$ that satisfies $$f_1+f_2\neq 0,\mbox{ ...
1
vote
2answers
238 views

Characteristic polynomial of Kronecker/tensor product

This was asked before on stackexchange but no answer was given. The question is the following: Let $A$ and $B$ be matrices in $GL(n)$ and $GL(m)$ respectively. Their tensor product $A\otimes B$ is ...
0
votes
0answers
115 views

Solution to system of polynomial equations

Suppose $A$ is a set of polynomials:$$P_1(x,y_1,\dots,y_n)=0,$$ $$P_2(x,y_1,\dots,y_n)=0,$$ $$\vdots$$ $$P_k(x,y_1,\dots,y_n)=0$$ is a system of equations with coefficients over $\mathbb{Z}$, and ...
4
votes
1answer
278 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} ...
1
vote
0answers
90 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 ...
12
votes
3answers
398 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}} ...
0
votes
0answers
41 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
0answers
82 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 ...
21
votes
3answers
875 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 ...
16
votes
1answer
944 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 ...
5
votes
1answer
208 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 ...
9
votes
2answers
275 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 ...
3
votes
1answer
195 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 ...
10
votes
3answers
500 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 ...
8
votes
1answer
202 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 ...
0
votes
0answers
126 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 ...
1
vote
1answer
204 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. ...
7
votes
3answers
344 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 ...
5
votes
2answers
185 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
0answers
95 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 ...
2
votes
1answer
199 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 ...
10
votes
1answer
270 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 ...
15
votes
3answers
616 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 ...
0
votes
1answer
197 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 ...
0
votes
1answer
63 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 ...
6
votes
1answer
144 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
1answer
96 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
0answers
57 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 ...