an: Zbl pre06399743
au: Ghorbani, A.; Nazemian, Z.
ti: On commutative rings with uniserial dimension.
la: EN
so: J. Algebra Appl. 14, No. 1, Article ID 1550008, 10 p. (2015).
py: 2015
dt: J
cc: *13A18 16P70 13E05 13E10 03E10
ut: uniserial module; uniserial dimension; valuation dimension
ab: Summary: In this paper, we define and study a valuation dimension for
commutative rings. The valuation dimension is a measure of how far a
commutative ring deviates from being valuation. It is shown that a ring $R$
with valuation dimension has finite uniform dimension. We prove that a ring
$R$ is Noetherian (respectively, Artinian) if and only if the ring $R
{\times} R$ has (respectively, finite) valuation dimension if and only if
$R$ has (respectively, finite) valuation dimension and all cyclic uniserial
modules are Noetherian (respectively, Artinian). We show that the class of
all rings of finite valuation dimension strictly lies between the class of
Artinian rings and the class of semi-perfect rings.