Centrally essential group algebras

2019

Beitr Algebra Geom

“…. c n ∈ C such that c n is a non-zero-divisor in R (resp., an invertible element in R) and (1) c n r n + c n−1 r n−1 + . .…”

Section: Rings Which Are Algebraic or Integral Over Their Centersmentioning

confidence: 99%

“…

MSC2010 datebase 16R99; 16D101 Centrally essential rings are studied, for example, in [1].

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mentioning

confidence: 99%

Rings with polynomial identity and centrally essential rings

Марков

2019

Beitr Algebra Geom

“…It is clear that all commutative rings are centrally essential. If Z 2 is the field of order 2 and Q 8 is the quaternion group of order 8, then the group ring Z 2 [Q 8 ] is an example of a non-commutative centrally essential ring [7]; in addition, Z 2 [Q 8 ] is a finite local ring.…”

Section: Introductionmentioning

confidence: 99%

Uniserial Noetherian centrally essential rings

Марков

Communications in Algebra

2019

It is proved that a ring A is a right or left Noetherian, right distributive centrally essential ring if and only if A = A 1 × · · · × A n , where each of the rings A i is either a commutative Dedekind domain or a uniserial Artinian centrally essential (not necessarily commutative) ring.V.T.Markov is supported by the Russian Foundation for Basic Research, project 17-01-00895-A. A.A.Tuganbaev is supported by Russian Scientific Foundation, project 16-11-10013.

“…A ring R with center C is said to be centrally essential if R C is an essential extension of the module C C , i.e., for every nonzero element r ∈ R, there exist two nonzero central elements x, y ∈ R with rx = y. Centrally essential rings are studied in many papers; e.g., see [5].…”

mentioning

confidence: 99%

“…There are many noncommutative centrally essential rings. For example, if F is the field Z/2Z and Q 8 is the quaternion group of order 8, then the group algebra FQ 8 is a finite noncommutative centrally essential ring; see [5].…”

mentioning

confidence: 99%

On Rings of Weak Global Dimension at Most One

2021

Mathematics

A ring R is of weak global dimension at most one if all submodules of flat R-modules are flat. A ring R is said to be arithmetical (resp., right distributive or left distributive) if the lattice of two-sided ideals (resp., right ideals or left ideals) of R is distributive. Jensen has proved earlier that a commutative ring R is a ring of weak global dimension at most one if and only if R is an arithmetical semiprime ring. A ring R is said to be centrally essential if either R is commutative or, for every noncentral element x∈R, there exist two nonzero central elements y,z∈R with xy=z. In Theorem 2 of our paper, we prove that a centrally essential ring R is of weak global dimension at most one if and only is R is a right or left distributive semiprime ring. We give examples that Theorem 2 is not true for arbitrary rings.