On the annihilator ideal of an inverse form: addendum

Let U be either the classical or quantized universal enveloping algebra of the Lie algebra [Formula: see text] extended over the field of fractions of the Cartan subalgebra. We suggest a PBW basis in U over the extended Cartan subalgebra diagonalizing the contravariant Shapovalov form on generic Verma module. The matrix coefficients of the form are calculated and the inverse form is explicitly constructed.

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Annihilator Ideal-Based Zero-Divisor Graphs Over Multiplication Modules

Communications in Algebra ◽

10.1080/00927872.2011.638353 ◽

2013 ◽

Vol 41 (3) ◽

pp. 1134-1148 ◽

Cited By ~ 1

Author(s):

Ghalandarzadeh ◽

S. Shirinkam ◽

P. Malakooti Rad

Keyword(s):

Zero Divisor ◽

Annihilator Ideal

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When the annihilator-ideal graph is planar or complete bipartite graph

Asian-European Journal of Mathematics ◽

10.1142/s1793557119500244 ◽

2019 ◽

Vol 12 (02) ◽

pp. 1950024

Author(s):

M. J. Nikmehr ◽

S. M. Hosseini

Keyword(s):

Commutative Ring ◽

Bipartite Graph ◽

Complete Bipartite Graph ◽

Simple Graph ◽

Annihilator Ideal ◽

Vertex Set

Let [Formula: see text] be a commutative ring with identity and [Formula: see text] be the set of ideals of [Formula: see text] with nonzero annihilator. The annihilator-ideal graph of [Formula: see text], denoted by [Formula: see text], is a simple graph with the vertex set [Formula: see text], and two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text]. In this paper, we present some results on the bipartite, complete bipartite, outer planar and unicyclic of the annihilator-ideal graphs of a commutative ring. Among other results, bipartite annihilator-ideal graphs of rings are characterized. Also, we investigate planarity of the annihilator-ideal graph and classify rings whose annihilator-ideal graph is planar.

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A Note on Quasi-Frobenius Rings and Ring Epimorphisms

Canadian Mathematical Bulletin ◽

10.4153/cmb-1969-036-9 ◽

1969 ◽

Vol 12 (3) ◽

pp. 287-292 ◽

Cited By ~ 5

Author(s):

H. H. Storrer

Keyword(s):

Unit Element ◽

Usual Condition ◽

Annihilator Ideal ◽

Frobenius Rings ◽

Weakened Form

In this note, we characterize quasi-Frobenius rings by a weakened form of the usual condition, that every ideal is an annihilator ideal.We then apply this result to pure rings in the sense of Cohn and to dominant rings, a concept arising in the study of ring epimorphisms. All rings considered have a unit element.

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Two notes on spectral synthesis for discrete Abelian groups

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100038950 ◽

1965 ◽

Vol 61 (3) ◽

pp. 617-620 ◽

Cited By ~ 17

Author(s):

R. J. Elliott

Keyword(s):

Fourier Transform ◽

Dual Space ◽

Fourier Transforms ◽

Spectral Synthesis ◽

Division Problem ◽

Exponential Functions ◽

Translation Invariant ◽

Annihilator Ideal ◽

The Fourier Transform ◽

Translation Invariant Subspace

Introduction. Spectral synthesis is the study of whether functions in a certain set, usually a translation invariant subspace (a variety), can be synthesized from certain simple functions, exponential monomials, which are contained in the set. This problem is transformed by considering the annihilator ideal in the dual space, and after taking the Fourier transform the problem becomes one of deciding whether a function is in a certain ideal, that is, we have a ‘division problem’. Because of this we must take into consideration the possibility of the Fourier transforms of functions having zeros of order greater than or equal to 1. This is why, in the original situation, we study whether varieties are generated by their exponential monomials, rather than just their exponential functions. This viewpoint of the problem as a division question, of course, perhaps throws light on why Wiener's Tauberian theorem works, and is implicit in the construction of Schwartz's and Malliavin's counter examples to spectral synthesis in L1(G) (cf. Rudin ((4))).

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