Since Buchberger introduced the theory of Gr obner bases in 1965 it has become an important tool in constructive algebra and, nowadays, Buchberger’s method is fundamental for many algorithms in the theory of polynomial ideals and algebraic geometry. Motivated by the results in polynomial rings a lot of possibilities to generalize the ideas to other types of rings have been investigated. The pe...

Let $R$ be a strongly $\mathbb {Z}^2$ -graded ring, and let $C$ bounded chain complex of finitely generated free -modules. The is $R_{(0,0)}$ -finitely dominated, or type $FP$ over , if it homotopy equivalent to projective We show that this happens only becomes acyclic after taking tensor product with certain eight rings formal power series, the graded analogues classical Novikov rings. This ex...

We study transcendency properties for graded field extension and give an application to valued field extensions. 1. Introduction. An important tool to study rings with valuation is the so-called associated graded ring construction: to a valuation ring R, we can associate a ring gr(R) graded by the valuation group. This ring is often easier to study, and one tries to lift properties back from gr...

Let $G$ be a group with identity $e$. $R$ $G$-graded commutative ring and $M$ graded $R$-module. In this paper, we introduce the concept of classical strongly 2-absorbing second submodules modules over rings. A number results concerning these classes their homogeneous components are given.

We obtain a complete structural characterization of Cohn-Leavitt algebras over no-exit objects as graded involutive algebras. Corollaries of this result include graph-theoretic conditions characterizing when a Leavitt path algebra is a directed union of (graded) matricial algebras over the underlying field and over the algebra of Laurent polynomials and when the monoid of isomorphism classes of...

In the past 15 years a study of “noncommutative projective geometry” has flourished. By using and generalizing techniques of commutative projective geometry, one can study certain noncommutative graded rings and obtain results for which no purely algebraic proof is known. For instance, noncommutative graded domains of quadratic growth, or “noncommutative curves,” have now been classified by geo...

We previously showed that the inverse limit of standard-graded polynomial rings with perfect (or semiperfect) coefficient field is a ring in an uncountable number variables. In this paper, we show result holds no hypothesis on field. also prove analogous for ultraproducts rings.

In this paper we prove analogs for the case of bigraded polynomial rings of theorems about regularity and saturation of ideals in simply graded polynomial rings.