<abstract><p>Let $ R be a G graded commutative ring and M $-graded $-module. The set of all second submodules is denoted by Spec_G^s(M), it called the spectrum $. We discuss rings with Noetherian prime spectrum. In addition, we introduce notion Zariski socle explore their properties. also investigate Spec^s_G(M) topology from viewpoint being space.</p></abstract>

Noetherian spaces are a topological concept that generalizes well quasiorderings. We explore applications to infinite-state verification problems, and show how this stimulated the search for infinite procedures à la Karp-Miller.

We give a theory on the space of arcs X∞ of a singular variety X, based on the finiteness property of the stable points of X∞. We also show how computations can be made in this theory. As a consequence, we obtain a process of determining invariants of X from its space of arcs X∞, and from its spaces of wedges (arcs in the space of arcs). More precisely, we give two concepts of regularity for th...

In this paper all coordinates in two variables over a Noetherian Q-domain of Krull dimension one are proved to be projectively tame. In order to do this, some results concerning projectively-tameness of polynomials in general are shown. Furthermore, we deduce that all automorphisms in two variables over a Noetherian reduced ring of dimension zero are tame.

For every variety of algebras Θ and every algebra H ∈ Θ we can consider an algebraic geometry in Θ over H. Algebras in Θ may be many sorted (not necessarily one sorted) algebras. A set of sorts Γ is fixed for each Θ. This theory can be applied to the variety of representations of groups over fixed commutative ring K with unit. We consider a representation as two sorted algebra (V, G), where V i...

In the paper we prove (Theorem 8.1) that there exists a continuum of non isomorphic simple modules over KF 2 , where F 2 is a free group with 2 generators (compare with [Ca] where a continuum of non isomorphic simple 2-generated groups is constructed). Using this fact we give an example of a non action type logically Noetherian representation (Section 9). In general, the topic of this paper is ...