Positively graded algebras are fairly natural objects which are arduous to be studied. In this article we query quotients of non-standard graded polynomial rings with combinatorial and commutative algebra methods.

Let $G$ be an abelian group with identity $e$. $R$ a $G$-graded commutative ring and $M$ graded $R$-module. In this paper we will obtain some results concerning the generalized 2-absorbing submodules their homogeneous components. Special attention has been paid, when rings are gr-Noetherian, to find extra properties of these submodules.

We introduce graded $${\mathbb {E}}_{\infty }$$ -rings and modules over them, study their properties. construct projective schemes associated to connective {N}}$$ -graded in spectral algebraic geometry. Under some finiteness conditions, we show that the $$\infty $$ -category of almost perfect quasi-coherent sheaves a scheme $$\text { {Proj}}\,(A)$$ -ring A can be described terms $${{\mathbb {Z}...

The duality theorem of Greenlees and May relating local cohomology with support on an ideal I and the left derived functors of I-adic completion [GM92] holds for rather general ideals in commutative rings. Here, simple formulas are provided for both local cohomology and derived functors of Z-graded completion, when I is a monomial ideal in the Z-graded polynomial ring k[x1, . . . , xn]. Greenle...

In this paper, we investigate certain graded-commutative rings which are related to the reciprocal plane compactification of coordinate ring a complement hyperplane arrangement. We give presentation these by generators and defining relations. This was used Holler I. Kriz calculate $\mathbb{Z}$-graded coefficients localizations ordinary $RO((\mathbb{Z}/p)^n)$-graded equivariant cohomology at giv...

The behaviour under coarsening functors of simple, entire, or reduced graded rings, free modules over principal superfluous monomorphisms and homological dimensions modules, as well adjoints degree restriction functors, are investigated.

A nilpotent quotient algorithm for graded Lie rings of prime characteristic is described. The algorithm has been implemented and applications have been made to the investigation of the associated Lie rings of Burnside groups. New results about Lie rings and Burnside groups are presented. These include detailed information on groups of exponent 5 and 7 and their associated Lie rings. Appeared: J...

There are remarkable relations between the graded homological dimensions and the ordinary homological dimensions. In this paper, we study the injective dimension of a complex of graded modules and derive its some properties. In particular, we define the $^*$dualizing complex for a graded ring and investigate its consequences.