Let Q be a (not necessarily unital) simple ring or algebra. A nonempty subset S of Q is said to have zero product if S2=0. We classify all maximal zero product subsets of Q by proving that the map ...R↦R∩LeftAnn(R) is a bijection from the set of all proper nonzero annihilator right ideals of Q onto the set of all maximal zero product subsets of Q. We also describe the relationship between the maximal zero product subsets of Q and the maximal inner ideals of its associated Lie algebra.
Suppose that R is an associative unital ring and that
$E=(E^0,E^1,r,s)$
is a directed graph. Using results from graded ring theory, we show that the associated Leavitt path algebra
$L_R(E)$
is simple ...if and only if R is simple,
$E^0$
has no nontrivial hereditary and saturated subset, and every cycle in E has an exit. We also give a complete description of the centre of a simple Leavitt path algebra.
We study the theory of diagonal reductions of matrices over simple Ore domains of finite stable range. We cover the cases of 2-simple rings of stable range 1, Ore domains and certain cases of Bezout ...domains.
Simplicity of Ore monoid rings Nystedt, Patrik; Öinert, Johan; Richter, Johan
Journal of algebra,
07/2019, Volume:
530
Journal Article
Peer reviewed
Open access
Given a non-associative unital ring R, a monoid G and a set π of additive maps R→R, we introduce the Ore monoid ring Rπ;G, and, in a special case, the differential monoid ring. We show that these ...structures generalize, in a natural way, not only the classical Ore extensions and differential polynomial rings, but also the constructions, introduced by Cojuhari, defined by so-called D-structures π. Moreover, for commutative monoids, we give necessary and sufficient conditions for differential monoid rings to be simple. We use this in a special case to obtain new and shorter proofs of classical simplicity results for differential polynomial rings in several variables previously obtained by Voskoglou and Malm by other means. We also give examples of new Ore-like structures defined by finite commutative monoids.
Rings with fine nilpotents Călugăreanu, Grigore; Zhou, Yiqiang
Annali dell'Università di Ferrara. Sezione 7. Scienze matematiche,
11/2021, Volume:
67, Issue:
2
Journal Article
Peer reviewed
A nonzero sum of a unit and a nilpotent element in a ring is called a fine element. This is a study of rings in which every nonzero nilpotent is fine, which we call
NF
rings.
We determine simplicity criteria in characteristics 0 and p for a ubiquitous class of iterated skew polynomial rings in two indeterminates over a base ring. One obstruction to simplicity is the ...possible existence of a canonical normal element z. In the case where this element exists we give simplicity criteria for the rings obtained by inverting z and the rings obtained by factoring out the ideal generated by z. The results are illustrated by numerous examples including higher quantized Weyl algebras and generalizations of some low-dimensional symplectic reflection algebras.
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