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Results tagged with ra.rings-and-algebras
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user 10503
Non-commutative rings and algebras, non-associative algebras, universal algebra and lattice theory, linear algebra, semigroups. For questions specific to commutative algebra (that is, rings that are assumed both associative and commutative), rather use the tag ac.commutative-algebra.
12
votes
Accepted
Ring with vanishing $K_0$
Question 1:
Does $K_0(R)=0$ imply that $R^m\approx R^n$ for some $m\neq n$?
Answer to Question 1:
Yes.
Choose any distinct natural numbers $m$ and $n$. Using square brackets to denote $K_0$ classes …
- 23k
3
votes
Accepted
Second summand to make projective module free
If you want something completely general, LSpice's comment is the answer.
For the special case of an ideal $P$ in the ring $R$ of integers of a number field (or more generally if $R$ is a Dedekind dom …
- 23k
10
votes
Accepted
Bass' stable range of $\mathbf Z[X]$
Sorry for putting this in a separate answer, but I think it will be cleaner this way.
The stable range of $\mathbb{Z}[x]$ is equal to 3.
I believe I now understand Vaserstein's intended argument …
- 63.9k
7
votes
Bass' stable range of $\mathbf Z[X]$
I have this from Vaserstein:
To show that ${\mathbb Z}[X]$ has stable range greater than 2, it suffices to find a quotient $A$ of ${\mathbb Z}[X]$ with $E_2(A)\neq SL_2(A)$.
For this, let $B$ be the …
- 63.9k
10
votes
Bass' stable range of $\mathbf Z[X]$
After spending considerable time trying to construct a counterexample, I turned to Google and found the book "Rings Related to Stable Range Conditions" by Huanyin Chen, which claims, on page 338, that …
- 63.9k
7
votes
Accepted
Milnor patching for general modules
Patching tends not to work well at the level of modules, especially if you want to preserve properties like finiteness of projective dimension. For most purposes, it's far better to work not with mod …
- 23k
4
votes
Accepted
Annihilator of minimal prime ideal in a commutative Noetherian ring
Put $P=(N:M)$.
Because $N$ is a maximal submodule, $P$ is a maximal ideal.
Because $M$ has finite length, there is some minimal $k$ such that $MP^k=MP^{k+1}$. By Nakayama, there exists $s\in 1 …
- 23k
4
votes
Properties of rings that have an elegant description in terms of the associated category of ...
If $R$ is an integral domain, then every ideal factors into primes if and only if every submodule of a projective module is projective.
- 23k
14
votes
Origin of exact sequences
There is a long exact sequence in a paper of Hurewicz in the Bulletin of the AMS, 1941. The paper is called "On Duality Theorems". According to Weibel's History of Homological Algebra, this is the f …
- 23k
5
votes
Accepted
Algebra structure $Tor(A,A)$
More generally, if $A,B,C,D $ are $k$-algebras, there's a multiplication
$$Tor_m^k(A,B)\otimes_k Tor_n^k(C,D)\rightarrow Tor^k_{m+n}(A\otimes_k C,B\otimes_k D)\qquad(1)$$
If you take $C=A$ and $B=D$, …
- 23k
4
votes
Accepted
Depth of polynomial ring $S=\Bbb{R}[x_1,x_2,x_3,...,x_n,...]$
In the same spirit as the answer to your earlier question, pick a bijection $\phi$ from the natural numbers to the rationals. For each real number $\alpha$, let $I_\alpha$ be the ideal generated by a …
CommunityBot
- 1
10
votes
Accepted
chain of prime ideals in polynomial ring $S=\Bbb{R}[x_1,x_2,...,x_n,...]$
Pick your favorite bijection $\phi$ from the natural numbers to the rational numbers. For each real number $\alpha$, let $I_\alpha$ be the ideal generated by all $x_n$ with $\phi(n)<\alpha$.
Then $I …
- 23k
1
vote
Accepted
Finite extension of a field
Suppose $k\subset Quot(A/p)$ is a finite extension of fields. Then every nonzero element $x$ of $A/p$ is algebraic over $k$ and so satisfies a minimal polynomial with non-zero constant term $a_0\in k …
- 23k
0
votes
zeros of a homogeneous polynomial
Over the field of three elements, you can take $\lambda=-1$. Over the field of five or seven elements, you can take $\lambda=1$. Over the field of eleven or thirteen elements, you can take $\lambda …
- 23k
3
votes
Accepted
ring with prescribed K group
Every abelian group $G$ is the class group of some Dedekind domain $R$ (theorem of Luther Claborn), so we have $K_0^{red}(R)= G$.
- 23k