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Questions tagged [ra.rings-and-algebras]

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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.

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Score of 2
0 answers
84 views

If $R$ and $S$ are two Morita equivalent rings, it is well known that their ideal lattices $\mathfrak{L}(R)$ and $\mathfrak{L}(S)$ are isomorphic. Namely, if we have a progenerator $(R,S)$-bimodule $P$...
Score of 1
0 answers
96 views

Let $(A,+)$ be an abelian group (the cases I care about are $\mathbb Z/m$ with $m\ge 3$, and $\mathbb Z$). Define $T\colon A\times A\to A$ by $$T(a,b)=\begin{cases}a+b,& a\neq 0,\\ 0,& a=0\end{...
Score of 13
1 answer
687 views

Let $(\mathrm{Solid}, \otimes^\square)$ denote the category of solid abelian groups. Is the functor $$\mathbb Z[[t]] \otimes_{\mathbb Z[t]}^\square - : \mathrm{Mod}_{\mathbb Z[t]}(\mathrm{Solid}) \to \...
Score of 2
1 answer
119 views

The Ado-Iwasawa theorem states that a finite dimensional Lie algebra $\mathfrak{g}$ over a field $k$ has a finite dimensional faithful representation $\mathfrak{g}\to \mathfrak{gl}_m(k)$. In ...
Score of 1
0 answers
45 views

Title: Intersection of a principal right ideal in an Ore localization with a quantum torus Let $$ A=\mathbb{K}\langle x^{\pm1},y^{\pm1},z^{\pm1}\rangle $$ be the quantum torus over a field $\mathbb{K}$...
Score of 1
0 answers
115 views

Gardam [1] disproved the Kaplansky unit conjecture by exhibiting a nontrivial unit $\alpha$ of support size 21 in $\mathbb{F}_2[P]$, where $P = \langle a,b \mid b^{-1}a^2b = a^{-2},\ a^{-1}b^2a = b^{-...
Score of 12
0 answers
645 views

Let $R$ be a two-sided noetherian ring (not necessarily commutative). Question: Is there a bijection between the simple left $R$- modules and the simple right $R$-modules? One might ask further ...
Score of 10
1 answer
1090 views

A rng—pronounced “rung” and henceforth spelt phonetically thus—is an algebraic structure that obeys all the axioms for a ring other than those involving a multiplicative identity element. For example,...
Score of 4
0 answers
254 views

I learned about the fact that the automorphisms of $k[x,y]$ for a field $k$ are all tame and in fact composition of triangular and linear automorphisms. Then I saw the Nagata's example as a wild ...
Score of 4
0 answers
136 views

I am confused about the following claims in chapter 1 of Erdmann's "Blocks of Tame Representation Type and Related Algebras". Let $k$ be an algebraically closed field of arbitrary ...
Score of 6
1 answer
208 views

Let $G$ be a group and $k$ a field. The Malcev completion $G \to \hat G$ over $k$ is (at least in this post) defined to be the group-like part $\mathbb{G}\widehat{k[G]}$ of the completed group ...
Score of 6
2 answers
280 views

What is an example of a finite dimensional Hopf algebra that does not admit a triangular, or even quasi-triangular structure? Looking at the classification of finite dimensional Hopf algebras up to ...
Score of 0
1 answer
113 views

Given an algebra $\mathcal{R}$ of subsets of a set $X$ (in the measure-theoretic sense) let $El(\mathcal{R})$ be the algebra of simple real functions on $\mathcal{R}$, that is, the linear combination ...
Score of 3
0 answers
168 views

In Section 10 of Facets of Module Theory over Semirings it was shown that an invertible module over a (commutative) semiring is Zariski-locally free. Therefore a local semiring must have trivial ...
Score of 7
1 answer
462 views

I'm looking for a reference for the following chain complex, which is a variation of the Koszul complex. Let $R$ be a commutative ring and $M$ an $R$-module with a surjective $R$-module homomorphism ...

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