Communications in Algebra

Cagliero, Leandro; Szechtman, Fernando

doi: 10.1080/00927872.2014.975352pmid: N/A

Let F be an algebraically closed field and consider the Lie algebra 𝔤 = ⟨ x ⟩ ⋉ 𝔞, where ad x acts diagonalizably on the abelian Lie algebra 𝔞. Refer to a 𝔤-module as admissible if [𝔤, 𝔤] acts via nilpotent operators on it, which is automatic if chr(F) = 0. In this article, we classify all indecomposable 𝔤-modules U which are admissible as well as uniserial, in the sense that U has a unique composition series.

Makedonskyi, Ievgen

doi: 10.1080/00927872.2013.865035pmid: N/A

Let 𝕂 be an algebraically closed field of characteristic zero and W n be the Lie algebra of all 𝕂-derivations of the polynomial ring R in n variables over 𝕂. It is proved that every Lie algebra of dimension n over 𝕂 can be isomorphically embedded in W n in such a way that any basis of its image (over 𝕂) is a basis of the free module W n over R.

Lavi, Noa

doi: 10.1080/00927872.2014.930473pmid: N/A

Let ⟨ K, ν ⟩ be a real closed valued field, and let S ⊆ K n be an open semialgebraic set. Using tools from model theory, we find an algebraic characterization of rational functions which admit, on S, only values in the valuation ring. We use this result to deduce a criterion for a rational function to be bounded on an open semialgebraic subset of some irreducible variety over a real closed field or over an ordered field which is dense in its real closure.

Bahturin, Yuri; Goze, Michel; Remm, Elisabeth

doi: 10.1080/00927872.2014.937535pmid: N/A

We classify, up to isomorphism, gradings by abelian groups on nilpotent filiform Lie algebras of nonzero rank over an algebraically closed field of characteristic 0. In case of rank 0, we describe conditions to obtain non trivial ℤ k -gradings.

Alhevaz, Abdollah; Kiani, Dariush

doi: 10.1080/00927872.2014.937536pmid: N/A

Hirano studied the quasi-Armendariz property of rings, and then this concept was generalized by some authors, defining quasi-Armendariz property for skew polynomial rings and monoid rings. In this article, we consider unified approach to the quasi-Armendariz property of skew power series rings and skew polynomial rings by considering the quasi-Armendariz condition in mixed extension ring [R; I][x; σ], introducing the concept of so-called (σ, I)-quasi Armendariz ring, where R is an associative ring equipped with an endomorphism σ and I is an σ-stable ideal of R. We study the ring-theoretical properties of (σ, I)-quasi Armendariz rings, and we obtain various necessary or sufficient conditions for a ring to be (σ, I)-quasi Armendariz. Constructing various examples, we classify how the (σ, I)-quasi Armendariz property behaves under various ring extensions. Furthermore, we show that a number of interesting properties of an (σ, I)-quasi Armendariz ring R such as reflexive and quasi-Baer property transfer to its mixed extension ring and vice versa. In this way, we extend the well-known results about quasi-Armendariz property in ordinary polynomial rings and skew polynomial rings for this class of mixed extensions. We pay also a particular attention to quasi-Gaussian rings.

Hirano, Yasuyuki; Matsuoka, Manabu; Poon, Edward; Tsutsui, Hisa

doi: 10.1080/00927872.2014.944264pmid: N/A

In this article we investigate: conditions under which a primitive idempotent in a ring is central; the set of right ideals eR where e is a primitive idempotent in a ring R; the set of primitive idempotent ideals of a ring; and idempotent elements in the ring of matrices.

Shen, Liang

doi: 10.1080/00927872.2014.955573pmid: N/A

A ring R is called right Johns if R is right noetherian and every right ideal of R is a right annihilator. R is called strongly right Johns if the matrix ring M n (R) is right Johns for each integer n ≥ 1. The Faith–Menal conjecture is an open conjecture on QF rings. It says that every strongly right Johns ring is QF. It is proved that the conjecture is true if every closed left ideal of the ring R is finitely generated. This result improves the known result that the conjecture is true if R is a left CS ring.

Evdokimov, Sergei; Kovács, István; Ponomarenko, Ilya

doi: 10.1080/00927872.2014.958848pmid: N/A

A finite group G is called a Schur group, if any Schur ring over G is associated in a natural way with a subgroup of Sym(G) that contains all right translations. Recently, the authors have completely identified the cyclic Schur groups. In this article, it is shown that any abelian Schur group belongs to one of several explicitly given families only. In particular, any noncyclic abelian Schur group of odd order is isomorphic to ℤ3 × ℤ3 k or ℤ3 × ℤ3 × ℤ p where k ≥ 1 and p is a prime. In addition, we prove that ℤ2 × ℤ2 × ℤ p is a Schur group for every prime p.

de Giovanni, Francesco; Ialenti, Roberto

doi: 10.1080/00927872.2014.958850pmid: N/A

It is proved that if G = AB is a soluble group with finite abelian section rank which is factorized by two mutually permutable finite-by-nilpotent subgroups A and B such that A′ and B′ are locally nilpotent, then also the normal closure ⟨ A′, B′ ⟩G is locally nilpotent and the subgroups A′ and B′ are ascendant in G.

Zhu, Haixing

doi: 10.1080/00927872.2014.966200pmid: N/A

In this article we show that the category of parachain complexes is equivalent to the category of Yetter–Drinfeld modules over the Pareigis's Hopf algebra.