## abstract algebra – How to precisely define \$b/a\$ if \$a mid b\$ in a general commutative ring?

Let $$R$$ be a commutative ring and let $$a,b in R$$. Suppose that $$a mid b$$, which means that there exists some $$k in R$$ such that $$kcdot a = b$$. I now often see expressions of the form $$b/a$$ in this context. ( For example the important lcm-gcd-formula $$gcd(a,b) = ab / lcm(a,b)$$ ). My question:

Is it enough to simply define $$b/a := k$$ or are there some caveats? (I think one problem here might be that $$b/a$$ does not have to be unique with this naive definition.)

Could you please explain this to me?

## rt.representation theory – Is it possible for the reduction modulo \$p\$ of an non-commutative semisimple algebra to be commutative?

Suppose that $$I, X_1, ldots, X_{d-1}$$ are $$n times n$$ matrices with integer entries whose $$mathbb{Z}$$-span is a subalgebra of $$mathrm{Mat}_n(mathbb{Z})$$. Suppose that, thought of as a subalgebra of $$mathrm{Mat}_n(mathbb{C})$$, this algebra is semisimple and non-commutative. Thus, by Wedderburn’s Theorem, it is isomorphic to a direct product of complete matrix algebras $$mathrm{Mat}_r(mathbb{C})$$, with $$r ge 2$$ for at least one factor.

It is possible that there exists a prime $$p$$ and a field $$K$$ of characteristic $$p$$ such that, regarding $$I, X_1, ldots, X_{d-1}$$ as elements of $$mathrm{Mat}_n(K)$$ by reduction modulo $$p$$, the subalgebra of $$mathrm{Mat}_n(K)$$ spanned by $$I, X_1, ldots, X_{d-1}$$ is commutative, and still of dimension $$d$$?

## matrices – Can all finite dimensional non commutative algebras be embedded into matrix rings?

Suppose I have a finite (non-)commutative ring $$R/k$$ (over a field $$k$$ of char $$0$$) with a linear “trace” function $$t: R to k$$. Can I find square matrices $$A_1,dots,A_n$$ (of some dimension $$r$$) so that I have an embedding $$f: R to M_r(k)$$ compatible with the trace functions on both sides?

One restriction I can see for the trace function on $$R$$ is that it should be invariant under cyclic permutations : $$t(a_1a_2dots a_n) = t(a_2dots a_na_1)$$. Is this the only restriction?

## commutative algebra – Properties of \$k[x(x-1)]_{langle x(x-1) rangle} subseteq k[x]_{langle x rangle}\$

Let $$k$$ be aa arbitrary field.

Let $$R=k(x(x-1))_{langle x(x-1) rangle}$$ and let $$S=k(x)_{langle x rangle}$$,
$$m=x(x-1)R$$, $$n=xS$$, $$k(m)=R/m$$, $$k(n)=S/n$$.

We have, $$mS = n$$ (since $$x-1$$ is invertible in $$S$$).

Now, $$k(m) cong k$$ and $$k(n) cong k$$, so ​$$k(m) cong k(n)$$.

I am trying to figure out if the residue field extension $$k(m) to k(n)$$ is finite-dimensional or not.

Question: Is $$k(m) to k(n)$$ finite-dimensional or not? and why?

Remarks:

(1) I think that it is infinite dimensional;
indeed, I know that $$A=k(x(x-1)) subseteq k(x)=frac{k(x(x-1))(T)}{langle T^2-T-x(x-1) rangle}=B$$ is not separable,
since $$(T^2-T-x(x-1))’=2T-1$$ evaluated at $$x$$, $$2x-1$$, is not a unit of $$k(x)$$.

$$B$$ is Noetherian+finitely generated $$A$$-algebra, hence $$B otimes_A B$$ is Noetherian, and so the kernel of $$B otimes_A B to B$$ is a finitely generated ideal. In the case separablity is equivalent to (formal) unramifiedness.
Therefore, being non-separable impies that $$A subseteq B$$ is not (formally) unramified,
so there exists a maximal ideal $$N$$ of $$B$$, such that $$A_{N cap A} subseteq B_N$$ is not unramified.

Now I am not sure for which maximal ideals $$N$$ of $$B$$, such localizations are not unramified.

(2) $$A subseteq B$$ is free with basis $${1,x}$$, hence flat, hence every such localization $$R subseteq S$$ is flat.

Any hints and comments are welcome!

## nt.number theory – What is this binary commutative operation on \$mathbb{Q_+}\$?

Write $$0inmathbb{N}$$. For $$ninmathbb{Q}_+$$, if $$n=prod_{iinmathbb{N}}p_i^{alpha_i}$$ is the prime factorization of $$n$$ and $$D_n:mathbb{N}→mathbb{N}$$ is $$D_n(i)=alpha_i$$, then for all $$a,bgeq1$$ we have

$$D_a+D_b=D_{ab}$$
$$min(D_a,D_b) = D_{gcd(a,b)}$$
$$max(D_a,D_b) = D_{text{lcm}(a,b)}$$

What is $$D_aD_b$$? Since $$D_a$$ and $$D_b$$ have finite support, it must be $$D_x$$ for some $$x$$.

## Let \$R\$ be a commutative ring and \$I\$ and ideal of \$R\$. How can I prove that \$(I^e)^c={ain Rmid (exists sin S),sain I}?

I already proved $$supseteq$$ but I can’t prove $$subseteq$$. This comes form Exercise V.4.9 from Algebra: Chapter 0.

## soft question – Latest “A Term of Commutative Algebra” by Altman and Kleiman?

(Moderator, please turn this question to a community-wiki. I’ll post my answer soon. TIA.)

Where can I find the latest revision of A term of Commutative Algebra by Allen B. ALTMAN and Steven L. KLEIMAN? Is my 2013 version ok?

It is hard to locate the latest one; many old revisions and pointers to them are randomly scattered across the web.

This free textbook is intended to be an update of, and an improvement to “A & M”, i.e. Introduction to Commutative Algebra by Atiyah and MacDonald.

## Is it true that the structure of a commutative ordered semiring is unique on a commutative ordered monoid?

Is it true that the structure of a commutative ordered semiring is unique on a commutative ordered monoid (i.e., the structure of the monoid and the order are consistent)? I am not asking about existence.

## commutative algebra – Reference book to study dimension theory.

I’m looking for a book to study Dimension theory, In particular I want a book(s) which covers the following topics:

1.Krull-dimension and Examples, Dimension of an integral extensions
2.Noether’s Normalization Lemma and its consequences.
3.Graded rings and modules, Hilbert functions and series, Hilbert’s Theorem.
4.Hilbert-Samuel functions and polynomials, System of parameters, Dimension Theorem.
5.Dimension of polynomial rings over noetherian rings.
6.Normal rings and their characterizations and properties. Finiteness of integral closure

Please suggest me a good book(s) which covers the above topics. (I’m planning to do self study, so if any of the book(s) is intended for self study it’s really good. Even if not, I hope I can do, so you don’t have to restrict to books intended for self study). Thanks.

## gr.group theory – Cancellation property for commutative monoid

Consider the monoid $$M=mathbb{N}times {0,1}$$ where $$(n,a)*(m,b):=(n+m, acdot b).$$
The unit element is $$e:=(0,1)$$. Note that this monoid is torsion free. Now consider the maps
$$g:(M,*,e)rightarrow (mathbb{N}, +,0), g(n,a)=n$$
and $$f: (mathbb{N}, +,0) rightarrow (M,*,e)$$ such that $$f(0)=e$$ and $$f(n)=(n,0)$$. Then we have $$fcirc g=id$$, but $$f(1)=(1,0)$$ is not cancellative as
\$\$ (1,0)(0,0)=(1,0)=(1,0)(0,1).\$