Let R be an associative ring with identity and M be a left R-module. In this paper, we define modules that have the property (δ- CE) ((δ-CEE)), these are modules that have a δ-supplement (ample δ- ...supplements) in every cofinite extension which are generalized version of modules that have the properties (CE) and (CEE) introduced in 6 and so a generalization of Zöschinger’s modules with the properties (E) and (EE) given in 23. We investigate various properties of these modules along with examples. In particular we prove these: (1) a module M has the property (δ-CEE) if and only if every submodule of M has the property (δ-CE); (2) direct summands of a module that has the property (δ-CE) also have the property (δ-CE); (3) each factor module of a module that has the property (δ-CE) also has the property (δ-CE) under a special condition; (4) every module with composition series has the property (δ- CE); (5) over a δ-V -ring a module M has the property (δ-CE) if and only if M is cofinitely injective; (6) a ring R is δ-semiperfect if and only if every left R-module has the property (δ-CE).
The statement of Theorem 2.1 in 4 is not correct. We give a counterexample to this statement and following to this error we give further corrections. We also offer two better substitutions for ...Proposition 2.9 and Theorem 2.11 in 4.
This paper describes properties of three certain classes of rings determined by conditions on idempotents and units, namely, the condition that any two generators of each principal right ideal are ...associated (UG rings), the condition that every principal right ideal is generated by a sum of a unit and an idempotent (P
r
), and the condition xy = 0 implies xsy = 0 for a sum of idempotent and unit s and any elements x, y of a ring (idun-semicommutative rings). It is proved that the class of all UG rings contains every local as well as every von Neumann regular ring, and the condition P
r
is satisfied by both semiperfect and regular rings. Both local and abelian regular rings are proved to be necessarily idun-semicommutative. For all three classes are presented some closure properties and illustrating examples.
In this article, we continue the study of strongly m-clean ring which we introduced in the paper “On m-clean and strongly m-clean rings” (Purkait et al. 2020). Mainly, we characterize strongly ...m-clean ring in terms of m-semiperfect ring. For that, we introduce the notion of m-semiperfect ring and establish some results on this ring. We prove that under certain conditions a ring is m-semiperfect if and only if it is strongly m-clean and orthogonally m-finite.
In this article, we continue the study of strongly m-clean ring which we introduced in the paper "On m-clean and strongly m-clean rings" (Purkait et al. 2020). Mainly, we characterize strongly ...m-clean ring in terms of m-semiperfect ring. For that, we introduce the notion of m-semiperfect ring and establish some results on this ring. We prove that under certain conditions a ring is m-semiperfect if and only if it is strongly m-clean and orthogonally m-finite.
Centrally essential rings Markov, Viktor T.; Tuganbaev, Askar A.
Discrete mathematics and applications,
06/2019, Volume:
29, Issue:
3
Journal Article
Peer reviewed
Open access
A centrally essential ring is a ring which is an essential extension of its center (we consider the ring as a module over its center). We give several examples of noncommutative centrally essential ...rings and describe some properties of centrally essential rings.
Symmetry on zero and idempotents Han, Juncheol; Lee, Chang Ik; Lee, Yang
Communications in algebra,
02/2023, Volume:
51, Issue:
2
Journal Article
Peer reviewed
Alghazzawi and Leroy studied the structure of subsets satisfying the properties of symmetric and commutatively closed, that is,
for
implies
and
for
implies
respectively, where S is a subset of a ring ...R. In this article we discuss the structure of rings which are symmetric on zero (resp., idempotents). Such rings are also called symmetric (resp., I-symmetric). We first prove that if a polynomial
over a symmetric ring is a unit then a
0
is a unit and a
i
is nilpotent for all
based on this result, we obtain that for a reduced ring R, the group of all units of the polynomial ring over R coincides with one of R, and that polynomial rings over I-symmetric rings are identity-symmetric. It is proved that for an abelian semiperfect ring R, R is I-symmetric if and only if the units in R form an Abelian group if and only if R is commutative. It is also proved that for an I-symmetric ring R, R is π-regular if and only if
is a commutative regular ring and J(R) is nil, where J(R) is the Jacobson radical of R.
The history of generalized "stacked bases" theorem origins from the result of Hill and Megibben on abelian groups. We extend this theorem for modules over semiperfect rings and as a consequence we ...show that for a submodule H of a projective module G over a semiperfect ring, the following conditions are equivalent:
there exists a decomposition
into a direct sum of indecomposable modules P
i
, such that
G/H is a direct sum of a family of modules, isomorphic to factor modules of principal indecomposable modules.
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