# Questions tagged [ra.rings-and-algebras]

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.

2,077
questions

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### Finiteness questions for enveloping algebras

Let $\mathfrak{g}$ be a Lie algebra over a field of characteristic $0$. The interesting case for these questions is when $\mathfrak{g}$ is infinite dimensional.
Is $U(\mathfrak{g})$ a coherent ...

**6**

votes

**0**answers

120 views

### Newer versions of Mahler's Lemma

I'm trying to find a way to numerically ensure that two constructible numbers are equal (this would be done by a computer).
The idea is to find a polynomial $p(x)$ that contains both numbers as roots ...

**4**

votes

**2**answers

182 views

### Complete reducibility and field extension

Let $\pi$ be a representation of a Lie algebra $L$ in a finite-dimensional linear space $V$ over the field $F$. Let $K$ be a field extension of $F$. Let $\pi_K=\pi\otimes K$ be the corresponding ...

**3**

votes

**1**answer

121 views

### A local ring with a unique minimal ideal

Let $R$ be a commutative ring with 1 such that it is local with a unique non-idempotent minimal ideal, that is, there are two ideals $I$ and $m$ of $R$ such that for each ideal $K$ of $R$ with $0\not=...

**4**

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**0**answers

80 views

### Row rank and column rank of matrix with entries in a commutative ring

Let $R$ be a unital commutative ring and $A\in M_{n\times m}(R)$. Under which appropriate invariant "rank" of modules discussed
in "Ranks of Modules"
one can say that the row rank of $A$ is ...

**0**

votes

**1**answer

127 views

### If two monic polynomials of $\mathbb{Z}_p[X]$ (p-adic integer coefficient) are relatively prime modulo p, then do they generate $\mathbb{Z}_p[X]$?

I'm currently reading a paper of Rene Schoof, and I got stuck in a line. And I'm trying to check the above sentence.
Although that seems to be elementary, I hope someone can give me a counterexample ...

**4**

votes

**1**answer

171 views

### List of Casimir elements of low dimensional Lie algebras

I am interested in explicit formulae for the Casimir elements (or "Casimir operators") of low-dimensional, real, non-Abelian Lie algebras (d=2,3, and possibly 4). I am wondering if there is any ...

**2**

votes

**1**answer

117 views

### Completeness of the ring of Witt vectors

Suppose that $S$ is a perfect ring of char $p$ and is t-adically complete for some non-zero-divisor $t$. Is the ring of Witt vectors $W(S)$ $[t]$-adically complete? If so can i get a proof please?

**2**

votes

**1**answer

275 views

### Motivation to study the order theory (ring theory)

I'm currently reading a paper of Georges Gras on the Reflection Principle.
The paper uses some theorems about orders (ring theory) from the book "Maximal Orders" by Reiner. I find the book interesting,...

**0**

votes

**1**answer

141 views

### Localization and containment in commutative ring

Let $R$ be a commutative ring with identity and $x, y $ be fixed elements of $R$ such that for each maximal ideal $m$ of $R$ we have $\langle \frac{x}{1_m}\rangle\subseteq\langle \frac{y}{1_m}\rangle$ ...

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100 views

### Question about Local Henselian Rings

I have a question regarding properties/characterizations of local Henselian rings exploited in M. Artin's article "On Isolated Rational Singularities of Surfaces":
Here the relevant excerpt:
Remark: ...

**2**

votes

**0**answers

57 views

### Isoartinian and isosimple modules

I'm reading this article by A. Facchini and Z. Nazemian, in wich they discuss modules with chain conditions up to isomorphism. A couple of the main concepts are the following:
Definition
We say that ...

**1**

vote

**1**answer

126 views

### A property for primitive idempotents

Let $R$ be a (commutative) ring (with identity). A nonzero idempotent $e\in R$ is called primitive idempotent, whenever it has no decomposition into $a+b$ where $a$ and $b$ are nonzero orthogonal ($...

**3**

votes

**0**answers

70 views

### Is a specific endomorphism of $A_1$ an automorphism?

Let $k$ be a field of characteristic zero, and let $A_1(k)$ be the first Weyl algebra, namely, the associative non-commutative $k$-algebra generated by $x$ and $y$ subject to the relation $yx-xy=1$.
...

**1**

vote

**1**answer

134 views

### Special idempotents in a commutative ring

Let $R$ be a commutative ring with $1$ and $e$ be an idempotent element of $R$ with the property that if $e=x+y$ (where $x, y\in R$), then there exists $r\in R$ such that either $e=rx$ or $e=ry$. ...