let $r$ be an associative ring with identity and $z^*(r)$ be its set of non-zero zero divisors.  the zero-divisor graph of $r$, denoted by $gamma(r)$, is the graph whose vertices are the non-zero  zero-divisors of  $r$, and two distinct vertices $r$ and $s$ are adjacent if and only if $rs=0$ or $sr=0$.  in this paper, we bring some results about undirected zero-divisor graph of a monoid ring ov...

Despite the abundance of "sandwich" complexes, in which two cyclic aromatic hydrocarbon ligands flank a metal center, this motif has not been extended to sheets of multiple metal atoms. We prepared and isolated two such compounds. In the first, three palladium centers form a planar triangular array, capped by chlorides, between two cycloheptatrienyl ligands. In the second, a pentapalladium shee...

The aromatic rings of the title compound, C(13)H(10)O(3)·H(2)O, are aligned at dihedral angles of 20.6 (1) and 40.8 (1)° with respect to the triangular C(ar-yl)-C(=O)-C(ar-yl) fragment. The hy-droxy groups are each hydrogen-bond donors to separate water mol-ecules, the water mol-ecule itself being hydrogen-bonded to one hy-droxy group and one carbonyl group. The water mol-ecule exists in an unu...

Infinite-dimensional Lie algebras are introduced, which are only partially graded, and are specified by indices lying on cyclotomic rings. They may be thought of as generalizations of the Onsager algebra, but unlike it, or its sl(n) generalizations, they are not subalgebras of the loop algebras associated with sl(n). In a particular interesting case associated with sl(3), their indices lie on t...

Let k be a fixed field of arbitrary characteristic and let ? basic connected Nakayama k-algebra without simple projective modules. In this article we prove that if V is an indecomposable finitely generated Gorenstein-projective left ?-module, then the versal deformation ring R(?,V) (in sense F. M. Bleher author) universal stable after taking syzygies. We also following result. ?=(?B0?) triangul...

We obtain several fundamental results on finite index ideals and additive subgroups of rings as well model-theoretic connected components rings, which concern generating in finitely many steps inside groups rings. Let R be any ring equipped with an arbitrary additional first order structure, A a set parameters. show that whenever H is A-definable, subgroup (R,+), then H+R⋅H contains two-sided i...

An element $u$ of a ring $R$ is called \textsl{unipotent} if $u-1$ is
 nilpotent. Two elements $a,b\in R$ are \textsl{unipotent equivalent}
 there exist unipotents $p,q\in such that $b=q^{-1}ap$. square
 matrices $A,B$ \textsl{strongly unipotent equivalent} there
 triangular $P,Q$ with $B=Q^{-1}AP$.
 In this paper, over commutative reduced rings, we characterize the mat...