Centrally essential endomorphism rings of abelian groups

Abstract

Let R be a ring and let J(R), C(R) be its Jacobson radical and center, correspondingly. If R is a centrally essential ring and the factor ring R/J(R) is commutative, then any minimal right ideal is contained in the center C(R). A right Artinian (or right Noetherian subdirectly indecomposable) centrally essential ring is a right and left Artinian local ring. We describe centrally essential Noetherian subdirectly indecomposable rings and centrally essential rings with subdirectly indecomposable center. We give examples of non-commutative subdirectly indecomposable, centrally essential rings.

Local Centrally Essential Subalgebras of Triangular Algebras

“…In [9], there is an example of a centrally essential ring R with 1 = 0 such that the factor ring of R with respect to the prime radical is not a PI ring. Abelian groups with centrally essential endomorphism rings are considered in [5].…”

“…In this paper, we consider local centrally essential subalgebras of the algebra T n (F) of all upper triangular matrices, where F is a field of characteristic = 2. In particular, such subalgebras are of interest, since, for F = Q, they are quasi-endomorphism algebras of strongly indecomposable torsionfree Abelian groups of finite rank n. Quasi-endomorphism algebras of all such groups are local matrix subalgebras in algebra M n (Q) of all matrices of order n over the field Q; e.g., see [4,Chapter I,§5]. We remark that the algebra QE is the quasi-endomorphism algebra of a strongly indecomposable torsion-free Abelian group of prime rank p if and only if QE is isomorphic to a local subalgebra of T p (Q).…”

“…Then we can take a matrix from Z(A c ) as the matrix A; see In [7, Proposition 2.5], it is proved that the external algebra Λ(V ) over a field F of characteristic = 2 is a centrally essential algebra if and only if the dimension of the space V is odd. By considering the regular matrix representation of the algebra Λ(V ), we obtain that for an odd positive integer n > 1, there exists a non-commutative centrally essential subalgebra of the algebra N 2 n (F); also see [5,Example 3.5]. Therefore, the minimal order of matrices of a non-commutative the external centrally essential algebra is equal to 8.…”

Centrally Essential Torsion-Free Rings of Finite Rank

“…In [9], 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 [5].…”

“…We denote by R the ring End A. In [5], it is proved that an Abelian torsion-free group A of finite rank can have noncommutative centrally essential endomorphism ring only if A is a reduced strongly indecomposable group. Following [2, Lemma 3.1], we assume that IA = A for some maximal right ideal of the ring R. It follows from [2] that pR ⊆ I for some prime integer p. By Theorem 1.3, the ring R is quasiinvariant.…”