All Questions
20 questions with no upvoted or accepted answers
9
votes
0
answers
460
views
Does the book "Algebra III" exist (within the Encyclopaedia of Mathematical Sciences series from Springer)?
Within the series "Encyclopaedia of Mathematical Sciences", as published by Springer, one finds the 8 volumes, namely,
the volumes I, II, IV, V, VI, VII, VIII, IX but zbMath has no listing ...
- 161
7
votes
0
answers
228
views
Terminology for vanishing of Hochschild homology with symmetric coefficients?
In a title or abstract for a paper, if I say "Hochschild cohomology of this algebra $A$ vanishes in degrees two and above" then
it should hopefully be understood by most readers as saying $H^n(A,M)=0$...
- 25.8k
7
votes
0
answers
296
views
A roadmap to learn about finite-dimensional commutative associative real or complex unital algebras
I've always been secretly fascinated with the rich structure and applications of finite-dimensional associative unital algebras over complete fields. In particular, I am very much interested in the ...
6
votes
0
answers
224
views
Book or survey on Dedekind-finite rings
I'm seeking a book or a survey providing an overview, as rich as possible, of the literature on Dedekind-finite (or von Neumann-finite) rings (let me recall that a unital ring $R$ is Dedekind-finite ...
- 10.5k
5
votes
0
answers
788
views
Rings such that torsion-free/flat/projective modules are flat/projective/free
While thinking about this question (and specifically YCor's remarks), I tried to remember what can be said about rings such that every torsion-free module is free, and I could not; and such things, ...
4
votes
0
answers
119
views
Adjoining new factors for primes in UFDs
It is well-known that if we pass from a UFD to a new ring where we have factored one of the primes, it does not need to stay a UFD. The classic example is passing from $\mathbb{Z}$ to $\mathbb{Z}[\...
- 18.7k
4
votes
0
answers
216
views
Characterizing atomicity in a commutative domain
In Proposition 1.1 of [Math. Proc. Cambridge Phil. Soc. 64 (1968), No. 2, 251-264], P.M. Cohn famously claimed (without proof) that a commutative domain is atomic if and only if it satisfies the ...
- 10.5k
4
votes
0
answers
98
views
$\lambda$-Decomposition for Connes' Cyclic Complex
Let $k$ be a field of characteristic zero, and $A$ be a commutative unital $k$-algebra. Then the cyclic homology of $A$ has a $\lambda$-decomposition:
$$HC_{n}(A)=HC_{n}^{(1)}(A)\oplus \cdots \oplus ...
- 661
4
votes
0
answers
67
views
Existence of infinitely many pairwise non-associate atoms in a ring of polynomials in $k$ variables over a Dedekind-finite unital ring
The following comes as a by-product of a more abstract result, and I'm essentially looking for a reference to it (or to something more general than it).
Corollary. Let $R$ be a non-trivial Dedekind-...
- 10.5k
4
votes
0
answers
429
views
Auslander-Reiten-Quivers of representation-finite algebras having different 3-dimensional forms
I am looking for references, where I can find (pictures of) connected Auslander-Reiten-Quivers of representation-finite $k$-algebras ($k$ is a (preferably, but not necessarily finite) field) with one ...
- 1,755
3
votes
0
answers
245
views
Quick proof of the first part of Kaplansky's Theorem on characterization of Noetherian domains with all maximal ideals principal
I have been reading section 12 of this paper "Elementary Divisors and Modules" by I. Kaplansky (https://www.ams.org/journals/tran/1949-066-02/S0002-9947-1949-0031470-3/S0002-9947-1949-...
- 739
3
votes
0
answers
97
views
Infinite-dimensional wild commutative algebras with non-trivial units
Let $A$ be an arbitrary (not necessarily finite-dimensional) associative algebra over an algebraically closed field $K$, and let $\mathrm{fin\,}A$ denote the category of finite-dimensional $A$-modules....
- 482
3
votes
0
answers
112
views
Indecomposablity in purely inseparable extensions
Let $k$ be a field of characteristic $p$ (e.g. the separable closure of $\mathbb{F}_p(t)$) and consider the extension $k(x)/k$ where $x^p\in k$ but $x$ does not. Consider a (finitely generated) ...
- 31
3
votes
0
answers
47
views
Counting the monic atoms $f$ in the semiring $\mathbf N[x]$ with $f(0)=1$, bounded coefficients, and degree $k$ (in the limit as $k \to \infty$)
Let $H$ be the multiplicative monoid of the (usual) semiring of polynomials in one variable $x$ with coefficients in $\mathbf N$. Given $\alpha, k \in \mathbf N$, denote by $\mathcal A_k(\alpha)$ the ...
- 10.5k
3
votes
0
answers
86
views
$\mathbb Z$-torsion for some quadratically presented Lie rings
$\newcommand{\Z}{\mathbb{Z}}$
I asked this question on MSE but no answer so far, so I'm also asking it here.
Let $L$ be a Lie ring (a Lie algebra over $\Z$) with generators $x_1,\dots,x_n$ and ...
- 8,524
2
votes
0
answers
95
views
Socle of a quotient of the ring of differential operators of a polynomial ring
I have been reading the following paper:
https://www.sciencedirect.com/science/article/pii/S002240491000263X
Proposition 2.4(ii) shows that if $\mathfrak D$ is a ring of $k$-linear differential ...
- 81
2
votes
0
answers
62
views
Extensions of an ideal-theoretic criterion for a monoid to be BF
Let $H$ be a multiplicatively written, commutative monoid. We denote by $H^\times$ the set of units (or invertible elements) of $H$, and by $\mathcal A(H)$ the set of atoms (or irreducible elements) ...
- 10.5k
2
votes
0
answers
270
views
Looking for a reference in commutative algebra
I need "I.G. Macdonald, Secondary representation of modules over a commutative ring, Sympos. Math. 11 (1973) 23–43." in my research, but it seems to be very old and rare. Does anyone know a site for ...
- 21
1
vote
0
answers
136
views
Representations of finite groups over commutative rings-question and reference request
In a textbook of representation theory I have encountered the following statement without proof:
Let $R$ be a commutative ring and $G$ a finite group. If $M$ is a simple $RG$-module then the ...
- 137
1
vote
0
answers
115
views
cohomlogy of Diagonal ring
Let $S=\bigoplus_{\underline n\in\mathbb N^r } S_{\underline n}$ be a standard multigraded ring over a local ring and M be a finitely generated $\mathbb N^r $-graded $S$-module. Let $M_{\Delta}=\...
- 1,713