let $r$ be a commutative ring with identity and let $m$ be an $r$-module. a proper submodule $p$ of $m$ is called strongly prime submodule if $(p + rx : m)ysubseteq p$ for $x, yin m$, implies that $xin p$ or $yin p$. in this paper, we study more properties of strongly prime submodules. it is shown that a finitely generated $r$-module $m$ is artinian if and only if $m$ is noetherian and every st...

For any field F , we give an example of a prime finitely generated F -algebra of GK dimension 2 for which GK dimension is not finitely partitive. For each m ∈ {0, 1, 2, 3, . . .}∪{∞}, we give an example of a prime finitely generated F -algebra having GK dimension 2 which has classical Krull dimension m; the algebra of GK dimension 2 which has infinite classical Krull dimension has the additiona...

We study skew inverse power series extensions R[[y−1; τ, δ]], where R is a noetherian ring equipped with an automorphism τ and a τ -derivation δ. We find that these extensions share many of the well known features of commutative power series rings. As an application of our analysis, we see that the iterated skew inverse power series rings corresponding to nth Weyl algebras are complete, local, ...

Krull dimension measures the depth of the spectrum Spec(R) of a commutative ring R. Since Spec(R) is a spectral space, Krull dimension can be defined for spectral spaces. Utilizing Stone duality, it can also be defined for distributive lattices. For an arbitrary topological space, the notion of Krull dimension is less useful. Isbell [23] remedied this by introducing the concept of graduated dim...

1 Introduction