Centrally essential torsion-free rings of finite rank
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Centrally essential torsion-free rings of finite rank O. V. Lyubimtsev1 · A. A. Tuganbaev2,3 Received: 6 July 2020 / Accepted: 2 September 2020 © The Managing Editors 2020
Abstract It is proved that centrally essential rings, whose additive groups of finite rank are torsion-free groups of finite rank, are quasi-invariant but not necessarily invariant. Torsion-free Abelian groups of finite rank with centrally essential endomorphism rings are faithful. Keywords Centrally essential ring · Quasi-invariant ring · Faithful Abelian group Mathematics Subject Classification 16R99 · 20K30
1 Introduction We consider only associative rings with non-zero identity elements. For a ring R, we denote by C(R), J (R) and N (R) the center, the Jacobson radical and the upper radical of the ring R, respectively. For Abelian groups, we use the additive notation.
The work of O.V. Lyubimtsev is done under the state assignment No 0729-2020-0055. A.A. Tuganbaev is supported by Russian Scientific Foundation, project 16-11-10013P.
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A. A. Tuganbaev [email protected] O. V. Lyubimtsev [email protected]
1
Nizhny Novgorod State University, Nizhny Novgorod, Russia
2
National Research University ‘MPEI’, Moscow, Russia
3
Lomonosov Moscow State University, Moscow, Russia
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Beitr Algebra Geom
1.1 Centrally essential rings A ring R is said to be centrally essential if for every non-zero element a ∈ R, there exist non-zero central elements x, y with ax = y.1 Centrally essential rings with non-zero identity elements are studied in Markov and Tuganbaev (2018, 2019a, b, c, 2020a, b, c). Every centrally essential semiprime ring with 1 = 0 is commutative; see Markov and Tuganbaev (2018, Proposition 3.3). Examples of non-commutative group algebras over fields are given in Markov and Tuganbaev (2018). In addition, the Grassman algebra over a three-dimensional vector space over the field of order 3 also is a finite non-commutative centrally essential ring; see (Markov and Tuganbaev 2019a). In Markov and Tuganbaev (2019c), there is an example of a centrally essential ring R whose factor ring with respect to the prime radical of R is not a PI ring. Abelian groups with centrally essential endomorphism rings are considered in Lyubimtsev and Tuganbaev (2020). 1.2 Torsion-free rings of finite rank and faithful Abelian groups A ring R is called a torsion-free ring of finite rank (a tffr ring) if the additive group (R, +) of R is an Abelian torsion-free group of finite rank. A ring R is said to be right invariant (resp., right quasi-invariant) if every right ideal (resp., maximal right ideal) of R is an ideal of R. The words such as “an invariant ring” (resp., ‘a quasi-invariant ring”) mean “a right and left invariant ring” (resp., “right and left quasi-invariant ring”). An Abelian group A is said to be faithful if I A = A for every proper right ideal I of the endomorphism ring End A of the group A. It is known that torsion-free groups of finite rank with commutative endomorphism rings are faithful (see Arnold 1982, Theorem 5.9). Faticoni got a s
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