Rigid transjective modules
Rigidity of generalized Verma modules
We prove that generalized Verma modules induced from generic Gelfand-Zetlin modules, and generalized Verma modules associated with Enright-complete modules, are rigid. Their Loewy lengths and quotients of the unique Loewy filtrations are calculated for the regular block of the corresponding category 𝒪(𝔭,Λ).
Ring elements as sums of units
In an Artinian ring R every element of R can be expressed as the sum of two units if and only if R/J(R) does not contain a summand isomorphic to the field with two elements. This result is used to describe those finite rings R for which Γ(R) contains a Hamiltonian cycle where Γ(R) is the (simple) graph defined on the elements of R with an edge between vertices r and s if and only if r - s is invertible. It is also shown that for an Artinian ring R the number of connected components of the graph...
Ring varieties closed under ideal sums
Ringe mit nichtkommutativer Addition I.
Ringe mit verallgemeinerter Kupischreihe.
Ringe mit verallgemeinerter Kupischreihe. II.
Ringel-Hall algebras of hereditary pure semisimple coalgebras
We define and investigate Ringel-Hall algebras of coalgebras (usually infinite-dimensional). We extend Ringel's results [Banach Center Publ. 26 (1990) and Adv. Math. 84 (1990)] from finite-dimensional algebras to infinite-dimensional coalgebras.
Ringepimorphismen und Morita-projektive Moduln über kommutativen Dedekind-Ringen.
Ring-like operations is pseudocomplemented semilattices
Ring-like operations are introduced in pseudocomplemented semilattices in such a way that in the case of Boolean pseudocomplemented semilattices one obtains the corresponding Boolean ring operations. Properties of these ring-like operations are derived and a characterization of Boolean pseudocomplemented semilattices in terms of these operations is given. Finally, ideals in the ring-like structures are defined and characterized.
Rings admitting torsion injective covers
Rings all of whose additive endomorphisms are left multiplications.
Rings consisting entirely of certain elements
We completely determine when a ring consists entirely of weak idempotents, units and nilpotents. We prove that such ring is exactly isomorphic to one of the following: a Boolean ring; ; where is a Boolean ring; local ring with nil Jacobson radical; or ; or the ring of a Morita context with zero pairings where the underlying rings are or .
Rings decomposed into direct sums of J-rings and nil rings.
Rings generalized by tripotents and nilpotents
We present new characterizations of the rings for which every element is a sum of two tripotents and a nilpotent that commute. These extend the results of Z. L. Ying, M. T. Koşan, Y. Zhou (2016) and Y. Zhou (2018).
Rings Graded By a Generalized Group
The aim of this paper is to consider the ringswhich can be graded by completely simple semigroups. We show that each G-graded ring has an orthonormal basis, where G is a completely simple semigroup. We prove that if I is a complete homogeneous ideal of a G-graded ring R, then R/I is a G-graded ring.We deduce a characterization of the maximal ideals of a G-graded ring which are homogeneous. We also prove that if R is a Noetherian graded ring, then each summand of it is also a Noetherian module..
Rings in Post algebras.
Rings in which elements are the sum of a nilpotent and a root of a fixed polynomial that commute
An element in a ring R with identity is said to be strongly nil clean if it is the sum of an idempotent and a nilpotent that commute, R is said to be strongly nil clean if every element of R is strongly nil clean. Let C(R) be the center of a ring R and g(x) be a fixed polynomial in C(R)[x]. Then R is said to be strongly g(x)-nil clean if every element in R is a sum of a nilpotent and a root of g(x) that commute. In this paper, we give some relations between strongly nil clean rings and strongly...
Rings in which every proper right ideal is maximal
Rings in which ideals are annihilators