Related papers: Multiplicative Lie-type derivations on alternative…

Let $R$ be a commutative Noetherian ring of dimension $d$ and $M$ a commutative cancellative torsion-free seminormal monoid. Then (1) Let $A$ be a ring of type $R[d,m,n]$ and $P$ be a projective $A[M]$-module of rank $r \geq max\{2,d+1\}$.…

Let \theta be an involution of the semisimple Lie algebra g and g=k+p be the associated Cartan decomposition. The nilpotent commuting variety of (g,\theta) consists in pairs of nilpotent elements (x,y) of p such that [x,y]=0. It is…

Here we investigate the property of effectivity for adjoint divisors. Among others, we prove the following results: (i) A normal projective variety $X$ with at most canonical singularities is uniruled if and only if for each very ample…

The aim of this paper is to introduce and study Lie algebras and Lie groups over noncommutative rings. For any Lie algebra $\gg$ sitting inside an associative algebra $A$ and any associative algebra $\FF$ we introduce and study the algebra…

We proved in previous work that all real nilpotent Lie algebras of dimension up to $10$ carrying an ad-invariant metric are nice. In this paper we show by constructing explicit examples that nonnice irreducible nilpotent Lie algebras…

A ring is called a commutator ring if every element is a sum of additive commutators. In this paper we give examples of such rings. In particular, we show that given any ring R, a right R-module N, and a set X, End_R(\bigoplus_X N) and…

We describe derivations of several important associative and Lie rings of infinite matrices over general rings of coefficients.

An associative ring $R$ with identity is left pseudo-morphic if for every $a$$\in$$R$, there exists $b$$\in$$R$ such that $Ra=l_R(b)$. If, in addition, $l_R(a)=Rb$, then $R$ is called left morphic. $R$ is morphic if it is both left and…

We introduce the notion of a graded integral element, prove the counterpart of the lying-over theorem on commutative algebra in the context of left commutative rngs, and use the Hu-Liu product to select a class of noncommutative rings.

When does a Noetherian commutative ring $R$ have uniform symbolic topologies on primes--read, when does there exist an integer $D>0$ such that the symbolic power $P^{(Dr)} \subseteq P^r$ for all prime ideals $P \subseteq R$ and all $r >0$?…

By a [$K$-]approximate subring of a ring we mean an additively symmetric subset $X$ such that $X \cdot X \cup (X + X)$ is covered by finitely many [resp.\ $K$] additive translates of $X$. We prove a structure theorem for finite approximate…

Let A and A' be two Cstar-algebras with identities I_A and I_A', respectively, and P_1 and P_2 = I_A - P_1 nontrivial projections in A. In this paper we study the characterization of multiplicative star-Lie-Jordan-type maps, where the…

We prove a strengthened form of a conjecture of Sun on a determinant attached to a binary quadratic form. Let $n>3$ and let $c,d\in\Z$. If $n$ is composite, then \[ \det\big[(i^2+cij+dj^2)^{n-2}\big]_{0\leq i,j\leq n-1}\equiv 0\pmod {n^2}…

Motivated by the combinatorial properties of products in Lie algebras, we investigate the subset of permutations that naturally appears when we write the long commutator $[x_1, x_2, ..., x_m]$ as a sum of associative monomials. We…

It is shown that every commutative arithmetic ring $R$ has $lambda$-dimension $ leq 3$. An example of a commutative Kaplansky ring with $ lambda$-dimension 3 is given. If $R$ satisfies an additional condition then $ lambda$-dim($R$)

Let $D$ be a subset of a finite commutative ring $R$ with identity. Let $f(x)\in R[x]$ be a polynomial of positive degree $d$. For integer $0\leq k \leq |D|$, we study the number $N_f(D,k,b)$ of $k$-subsets $S\subseteq D$ such that…

We conjecture that if $G$ is a simple compact Lie group with trivial center, then every $d$-variable non-constant word map with coefficients in $G$ defines a non-constant function on $G^d$. We prove the conjecture for $A_r$, $B_r$, $E_6$,…

We introduce the concept of a triangular representation of a Lie algebra, give a counterpart of Ado's theorem, and discuss $2$-irreducible triangular modules over a nonreductive Lie algebra.

A prime ring $R$ with extended centroid $C$ is said to be exceptional if both $\text{\rm char}\,R=2$ and $\dim_CRC=4$. Herstein characterized additive subgroups $A$ of a nonexceptional simple ring $R$ satisfying $\big[A, [R,…

Let $n$ and $s$ be fixed integers such that $n\geq 2$ and $1\leq s\leq \frac{n}{2}$. Let $M_n(\mathbb{K})$ be the ring of all $n\times n$ matrices over a field $\mathbb{K}$. If a map $\delta:M_n(\mathbb{K})\rightarrow M_n(\mathbb{K})$…