It is proved that a finite-dimensional centrally essential algebra R is a centrally essential ring if and only if the formal power series ring
is centrally essential, if and only if the formal ...Laurent series ring
is centrally essential. There exists a centrally essential ring R such that the ring
with respect to the Jacobson radical is not a PI ring; consequently, the ring
with respect to the prime radical also is not a PI ring (in particular, the rings
and
are not commutative).
Rings essential over their centers Markov, Victor T.; Tuganbaev, Askar A.
Communications in algebra,
04/2019, Volume:
47, Issue:
4
Journal Article
Peer reviewed
A ring R with center C is said to be centrally essential if the module R
C
is an essential extension of the module C
C
. We describe centrally essential exterior algebras of finitely generated free ...modules over not necessary commutative rings and study properties of semi-Artinian centrally essential rings.
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.
On serial rings Tuganbaev, Askar A.
Discrete mathematics and applications,
04/2017, Volume:
27, Issue:
2
Journal Article
Peer reviewed
Let A be a ring such that all maximal indecomposable factor rings
of
are serial rings. Then every square matrix over
is diagonalizable. In addition, if all the rings
are Bezout rings, then every ...rectangular matrix over
is diagonalizable. If
is an automorphism of the ring
, then the skew Laurent series ring
((
,
)) is a serial ring if and only if
is a serial Artinian ring.
Centrally essential rings were defined earlier for associative unital rings; in this paper, we define them for rings which are not necessarily associative or unital. In this case, it is proved that ...centrally essential semiprime rings are commutative. It is proved that all idempotents of a centrally essential alternative ring are central. Several examples of non-commutative centrally essential rings are provided, some properties of centrally essential rings are described.
