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Search: MSC category 16W50 ( Graded rings and modules )

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1. CMB Online first

Dehghani-Zadeh, Fatemeh
Artinianness of Composed Graded Local Cohomology Modules
Let $R=\bigoplus_{n\geq0}R_{n}$ be a graded Noetherian ring with local base ring $(R_{0}, \mathfrak{m}_{0})$ and let $R_{+}=\bigoplus_{n\gt 0}R_{n}$, $M$ and $N$ be finitely generated graded $R$-modules and $\mathfrak{a}=\mathfrak{a}_{0}+R_{+}$ an ideal of $R$. We show that $H^{j}_{\mathfrak{b}_{0}}(H^{i}_{\mathfrak{a}}(M,N))$ and $H^{i}_{\mathfrak{a}}(M, N)/\mathfrak{b}_{0}H^{i}_{\mathfrak{a}}(M,N)$ are Artinian for some $i^{,}s$ and $j^{,}s$ with a specified property, where $\mathfrak{b}_{o}$ is an ideal of $R_{0}$ such that $\mathfrak{a}_{0}+\mathfrak{b}_{0}$ is an $\mathfrak{m}_{0}$-primary ideal.

Keywords:generalized local cohomology, Artinian, graded module
Categories:13D45, 13E10, 16W50

2. CMB 2011 (vol 55 pp. 271)

Di Vincenzo, M. Onofrio; Nardozza, Vincenzo
On the Existence of the Graded Exponent for Finite Dimensional $\mathbb{Z}_p$-graded Algebras
Let $F$ be an algebraically closed field of characteristic zero, and let $A$ be an associative unitary $F$-algebra graded by a group of prime order. We prove that if $A$ is finite dimensional then the graded exponent of $A$ exists and is an integer.

Keywords:exponent, polynomial identities, graded algebras
Categories:16R50, 16R10, 16W50

3. CMB 2011 (vol 55 pp. 208)

Valenti, Angela; Zaicev, Mikhail
Abelian Gradings on Upper Block Triangular Matrices
Let $G$ be an arbitrary finite abelian group. We describe all possible $G$-gradings on upper block triangular matrix algebras over an algebraically closed field of characteristic zero.

Keywords:gradings, upper block triangular matrices

4. CMB 2008 (vol 51 pp. 460)

Smoktunowicz, Agata
On Primitive Ideals in Graded Rings
Let $R=\bigoplus_{i=1}^{\infty}R_{i}$ be a graded nil ring. It is shown that primitive ideals in $R$ are homogeneous. Let $A=\bigoplus_{i=1}^{\infty}A_{i}$ be a graded non-PI just-infinite dimensional algebra and let $I$ be a prime ideal in $A$. It is shown that either $I=\{0\}$ or $I=A$. Moreover, $A$ is either primitive or Jacobson radical.

Categories:16D60, 16W50

5. CMB 2002 (vol 45 pp. 451)

Allison, Bruce; Smirnov, Oleg
Coordinatization Theorems For Graded Algebras
In this paper we study simple associative algebras with finite $\mathbb{Z}$-gradings. This is done using a simple algebra $F_g$ that has been constructed in Morita theory from a bilinear form $g\colon U\times V\to A$ over a simple algebra $A$. We show that finite $\mathbb{Z}$-gradings on $F_g$ are in one to one correspondence with certain decompositions of the pair $(U,V)$. We also show that any simple algebra $R$ with finite $\mathbb{Z}$-grading is graded isomorphic to $F_g$ for some bilinear from $g\colon U\times V \to A$, where the grading on $F_g$ is determined by a decomposition of $(U,V)$ and the coordinate algebra $A$ is chosen as a simple ideal of the zero component $R_0$ of $R$. In order to prove these results we first prove similar results for simple algebras with Peirce gradings.


6. CMB 2002 (vol 45 pp. 711)

Yoshii, Yoji
Classification of Quantum Tori with Involution
Quantum tori with graded involution appear as coordinate algebras of extended affine Lie algebras of type $\rmA_1$, $\rmC$ and $\BC$. We classify them in the category of algebras with involution. From this, we obtain precise information on the root systems of extended affine Lie algebras of type $\rmC$.


7. CMB 2002 (vol 45 pp. 499)

Bahturin, Yu. A.; Zaicev, M. V.
Group Gradings on Matrix Algebras
Let $\Phi$ be an algebraically closed field of characteristic zero, $G$ a finite, not necessarily abelian, group. Given a $G$-grading on the full matrix algebra $A = M_n(\Phi)$, we decompose $A$ as the tensor product of graded subalgebras $A = B\otimes C$, $B\cong M_p (\Phi)$ being a graded division algebra, while the grading of $C\cong M_q (\Phi)$ is determined by that of the vector space $\Phi^n$. Now the grading of $A$ is recovered from those of $A$ and $B$ using a canonical ``induction'' procedure.


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