Let L be a complete lattice. We introduce and characterize the prime L-submodules of a unitary module over a commutative ring with identity. Finally, we investigate the Zariski topology on the prime L-Spectrum of a unitary module, consisting of the collection of all prime L-submodules, and prove that for L-top modules the Zariski topology on L-Spec(M) exists. © 2007 Elsevier B.V. All rights res...

Let $f \colon X \to X$ be a surjective endomorphism of normal projective surface. When $\operatorname{deg} f \geq 2$, applying an (iteration of) $f$-equivariant minimal model program (EMMP), we determine the geometric structure $X$. Using this, extend second author's result to singular surfaces extent that either $X$ has $f$-invariant non-constant rational function, or $f$ infinitely many Zaris...

We consider the Zariski space of all places of an algebraic function field F |K of arbitrary characteristic and investigate its structure by means of its patch topology. We show that certain sets of places with nice properties (e.g., prime divisors, places of maximal rank, zerodimensional discrete places) lie dense in this topology. Further, we give several equivalent characterizations of field...

Let $R$ be an associative ring and let $M$ be a left $R$-module.Let $Spec_{R}(M)$ be the collection of all prime submodules of $M$ (equipped with classical Zariski topology). There is a conjecture which says that every irreducible closed subset of $Spec_{R}(M)$ has a generic point. In this article we give an affirmative answer to this conjecture and show that if $M$ has a Noetherian spectrum, t...

We introduce the notion of a Hu-Liu prime ideal in the context of left commutative rngs, and establish the contravariant functor from the category of left commutative rngs into the category of topological spaces. It is well known that new points must be introduced in order to expand algebraic geometry over algebraically closed fields into Grothendieck’s scheme theory over commutative rings. We ...

The prime k-spectrum Speck(R) of a k-semiring R will be introduced. It will be proven that it is a topological space, and some properties of this space will be investigated. Connections between the topological properties of Speck(R) and possible algebraic properties of the k-semiring R will be established.