Narrow Results

We continue the study of nil-Armendariz rings, initiated by Antoine, and Armendariz rings. We first examine a kind of ring coproduct constructed by Antoine for which the Armendariz, nil-Armendariz, and weak Armendariz properties are equivalent. Such a ring has an important role in the study of Armendariz ring property and near-related ring properties. We next prove an Antoine's result in relation with the ring coproduct by means of a simpler direct method. In the proof we can observe the concrete shapes of coefficients of zero-dividing polynomials. We next observe the structure of nil-Armendariz rings via the upper nilradicals. It is also shown that a ring R is Armendariz if and only if R is nil-Armendariz if and only if R is weak Armendariz, when R is a von Neumann regular ring.

This paper is devoted to the so-called complete Leibniz algebras. We construct some complete Leibniz algebras with complete radical and prove that the direct sum of complete Leibniz algebras is also complete. It is known that a Lie algebra with a complete ideal is split. We discuss the analogs of this result for the Leibniz algebras and show that it is true for some special classes of Leibniz algebras. Finally, we consider derivations of Leibniz algebras and present some classes of Leibniz algebras which are not complete, since they admit outer derivation.

In this paper, we determine derivations of the nilradicals of Borel subalgebras in Kac–Moody algebras (and contragredient Lie algebras) over any field of characteristic 0. The result solves a conjecture posed by Moody 30 years ago which generalizes a result by Kostant for finite-type simple Lie algebras. We also determine automorphisms of those subalgebras in symmetrizable Kac–Moody algebras.

Recently, Xiang and Ouyang defined a (commutative unital) ring $R$ to be Nil${}_{*}$-coherent if each finitely generated ideal of $R$ that is contained in Nil$(R)$ is a finitely presented $R$-module. We define and study Nil${}_{*}$-coherent modules and special Nil${}_{*}$-coherent modules over any ring. These properties are characterized and their basic properties are established. Any coherent ring is a special Nil${}_{*}$-coherent ring and any special Nil${}_{*}$-coherent ring is a Nil${}_{*}$-coherent ring, but neither of these statements has a valid converse. Any reduced ring is a special Nil${}_{*}$-coherent ring (regardless of whether it is coherent). Several examples of Nil${}_{*}$-coherent rings that are not special Nil${}_{*}$-coherent rings are obtained as byproducts of our study of the transfer of the Nil${}_{*}$-coherent and the special Nil${}_{*}$-coherent properties to trivial ring extensions and amalgamated algebras.

A formula for the number of gradations, up to equivalence, of cyclic rings by cancellative monoids is given. As an application, the nil and Jacobson radicals of cyclic rings are shown to be homogeneous.

In this work, we consider extensions of solvable Lie algebras with naturally graded filiform nilradicals. Note that there exist two naturally graded filiform Lie algebras $n_{n,1}$ and $Q_{2n}.$ We find all one-dimensional extensions of solvable Lie algebras with nilradical $n_{n,1}$. We prove that there exists a unique non-split central extension of solvable Lie algebras with nilradical $n_{n,1}$ of maximal codimension. Moreover, all one-dimensional extensions of solvable Lie algebras with nilradical $n_{n,1}$ whose codimension is equal to one are found and we compared these solvable algebras with the solvable algebras with nilradicals that are one-dimensional central extension of algebra $n_{n,1}$.

The paper is devoted to the so-called complete Leibniz algebras. It is known that a Lie algebra with a complete ideal is split. We will prove that this result is valid for Leibniz algebras whose complete ideal is a solvable algebra such that the codimension of nilradical is equal to the number of generators of the nilradical.

A corrected and completed list of six dimensional real Lie algebras with five dimensional nilradical is presented. Their invariants for the coadjoint representation are computed and some results on the invariants of solvable Lie algebras in arbitrary dimension whose nilradical has codimension one are also given. Specifically, it is shown that any rank one solvable Lie algebra of dimension n without invariants determines a family of (n+2k)-dimensional algebras with the same property.

In this paper we study some properties of the Hurwitz series ring HA over a commutative ring A, such as the nilradical of HA and the chain condition on its annihilators. We provide an example showing that the last property does not pass from A to HA. A strongly Hopfian ring is a ring satisfying the chain condition on some type of annihilators. We give a large class of strongly Hopfian rings A such that HA are not strongly Hopfian.

Marks called a ring NI if the upper nilradical contains all nilpotent elements. Smoktunowicz constructed an NI ring over which the polynomial ring is not NI. We study some conditions under which polynomial rings are NI. In the process we call a ring Rpolynomial-NI if R[Y] is NI for any finite set Y of commuting indeterminates over R. We show that R is polynomial-NI if and only if R[X] is polynomial-NI for any (possibly infinite) set X of commuting indeterminates over R. It is also shown that R is polynomial-NI if and only if R[x] is NI when R is a ring such that every finite subset of R generates a subring of bounded index of nilpotency, where x is an indeterminate over R.