## What is/are Graded Rings?

Graded Rings - The rings we encounter are graded rings of the form K [ U ] Γ where Γ is a product of general linear groups over a field K of characteristic zero, and U is a finite dimensional rational representation of Γ.^{[1]}org/1998/Math/MathML" alttext="upper Q">

^{[2]}Also we study $$\mathrm{Cay}(R, S'')$$ as another Cayley graph over graded rings and obtain relations between this graph and total, cototal and counit graphs.

^{[3]}In this article, we consider the structure of graded rings, not necessarily commutative nor with unity, and study the graded weakly prime ideals.

^{[4]}In this article, we study the characterizations on this new class of graded ideals, and investigate the behaviour of graded uniformly pr-ideals in graded factor rings and in direct product of graded rings.

^{[5]}We are considering the corresponding concepts in bigraded rings.

^{[6]}In this paper, we discuss some open problems of non-commutative algebra and non-commutative algebraic geometry from the approach of skew PBW extensions and semi-graded rings.

^{[7]}We give generators and relations for the graded rings of Hermitian modular forms of degree two over the rings of integers in $${\mathbb {Q}}(\sqrt{-7})$$ Q ( - 7 ) and $${\mathbb {Q}}(\sqrt{-11})$$ Q ( - 11 ).

^{[8]}However, the behavior of this new notion has not been enough investigated for the principal algebraic constructions as polynomial rings, matrix rings, localizations, filtered–graded rings, skew PBW extensions, etc.

^{[9]}We present a generic construction of efficient USS-ADL-QANIZK for diverse vector spaces (DVS) over graded rings, of which linear subspaces over bilinear groups are specific instantiations.

^{[10]}The algebraic Cuntz-Pimsner rings are naturally $\mathbb{Z}$-graded rings that generalize both Leavitt path algebras and unperforated $\mathbb{Z}$-graded Steinberg algebras.

^{[11]}In this article, we compute the set of point modules of finitely semi-graded rings.

^{[12]}We compute generators and relations for the graded rings of paramodular forms of degree two and levels 5 and 7.

^{[13]}Moreover, the point modules and the point functor are introduced for finitely semi-graded rings.

^{[14]}The algebraic Cuntz-Pimsner rings are naturally Z-graded rings that generalize both Leavitt path algebras and unperforated Z-graded Steinberg algebras.

^{[15]}Moreover, we show that every super-biderivation of these rings is a biderivation, and a question is posed on super-biderivations of graded rings.

^{[16]}We specify exterior generators for $\pi_* THH(MU) = \pi_*(MU) \otimes E(\lambda'_n \mid n\ge1)$ and $\pi_* THH(BP) = \pi_*(BP) \otimes E(\lambda_n \mid n\ge1)$, and calculate the action of the $\sigma$-operator on these graded rings.

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## Associated Graded Rings

We also give a few results relating the (◇) property of a local ring with that of its associated graded rings, and construct a series of examples.^{[1]}We go on to prove a parametrisation of the indecomposable objects in all interpolating partition categories for non-zero interpolation parameters via a system of finite groups, which we associate to any partition category, and which we also use to describe the associated graded rings of the Grothendieck rings of these interpolation categories.

^{[2]}We show that certain homological regularity properties of graded connected algebras, such as being AS-Gorenstein or AS-Cohen-Macaulay, can be tested by passing to associated graded rings.

^{[3]}Moreover, we establish a relationship between the vanishing of Hilbert coefficients and the depth of associated graded rings with respect to parameter ideals in the case of small regularity.

^{[4]}

## Strongly Graded Rings

Abrams, Invariant basis number and types for strongly graded rings, J.^{[1]}We provide a characterization of graded von Neumann regular rings involving the recently introduced class of nearly epsilon-strongly graded rings.

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