1. CMB 2005 (vol 48 pp. 275)
|Krull Dimension of Injective Modules Over Commutative Noetherian Rings |
Let $R$ be a commutative Noetherian integral domain with field of fractions $Q$. Generalizing a forty-year-old theorem of E. Matlis, we prove that the $R$-module $Q/R$ (or $Q$) has Krull dimension if and only if $R$ is semilocal and one-dimensional. Moreover, if $X$ is an injective module over a commutative Noetherian ring such that $X$ has Krull dimension, then the Krull dimension of $X$ is at most $1$.
Categories:13E05, 16D50, 16P60
2. CMB 2005 (vol 48 pp. 317)
|On Pseudo-Frobenius Rings |
It is proved here that a ring $R$ is right pseudo-Frobenius if and only if $R $ is a right Kasch ring such that the second right singular ideal is injective.
3. CMB 1998 (vol 41 pp. 261)
|A simple ring over which proper cyclics are continuous is a $\PCI$-ring |
It is shown that simple rings over which proper cyclic right modules are continuous coincide with simple right $\PCI$-rings, introduced by Faith.
Keywords:Simple rings, $\PCI$-rings, $\PCQI$-rings, continuous modules,, quasi-continuous modules