  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. The work of Oleg Lyubimtsev is supported by Ministry of
Education and Science of the Russian Federation, project FSWR-2023-0034. The
study of Askar Tuganbaev is supported by grant of Russian Science Foundation
(=RSF), project 22-11-00052, https://rscf.ru/en/project/22-11-00052.
