Annihilator Ideals Research Articles

Algebraic quotient modules and subgroup depth

In Kadison J Pure Appl Alg 218:367–380, (2014) it was shown that subgroup depth may be computed from the permutation module of the left or right cosets: this holds more generally for a Hopf subalgebra, from which we note in this paper that finite depth of a Hopf subalgebra $$R \subseteq H$$ is equivalent to the $$H$$ -module coalgebra $$Q = H/R^+H$$ representing an algebraic element in the Green ring of $$H$$ or $$R$$ . This approach shows that subgroup depth and the subgroup depth of the corefree quotient lie in the same closed interval of length one. We also establish a previous claim that the problem of determining if $$R$$ has finite depth in $$H$$ is equivalent to determining if $$H$$ has finite depth in its smash product $$Q^* \# H$$ . A necessary condition is obtained for finite depth from stabilization of a descending chain of annihilator ideals of tensor powers of $$Q$$ . As an application of these topics to a centerless finite group $$G$$ , we prove that the minimum depth of its group $$\mathbb {C}\,$$ -algebra in the Drinfeld double $$D(G)$$ is an odd integer, which determines the least tensor power of the adjoint representation $$Q$$ that is a faithful $$\mathbb {C}\,G$$ -module.

A new class of non-semiprime quasi-Armendariz rings

Hirano [On annihilator ideals of a polynomial ring over a noncommutative ring, J. Pure Appl. Algebra, 168 (2002), 45–52] studied relations between the set of annihilators in a ring R and the set of annihilators in a polynomial extension R[x] and introduced quasi-Armendariz rings. In this paper, we give a sufficient condition for a ring R and a monoid M such that the monoid ring R[M] is quasi-Armendariz. As a consequence we show that if R is a right APP-ring, then R[x]=(xn) and hence the trivial extension T(R,R) are quasi-Armendariz. They allow the construction of rings with a non-zero nilpotent ideal of arbitrary index of nilpotency which are quasi-Armendariz.

Green's relations in Partial \(\Gamma\)-Semirings

A \(\Gamma\)-so-ring is a structure possessing a natural partial ordering, an infinitary partial addition and a ternary multiplication, subject to a set of axioms. The partial functions under disjoint-domain sums and functional composition is a \(\Gamma\)-so-ring. In this paper we introduce the notions of dense and annihilator ideals of \(\Gamma\)-so-rings and obtain the Green's relations in Partial \(\Gamma\) semirings.

Some new classes of topological spaces and annihilator ideals

By a characterization of semiprime SA-rings by Birkenmeier, Ghirati and Taherifar in [4, Theorem 4.4], and by the topological characterization of C(X) as a Baer-ring by Stone and Nakano in [11, Theorem 3.25], it is easy to see that C(X) is an SA-ring (resp., IN-ring) if and only if X is an extremally disconnected space. This result motivates the following questions: Question (1): What is X if for any two ideals I and J of C(X) which are generated by two subsets of idempotents, Ann(I)+Ann(J)=Ann(I∩J)? Question (2): When does for any ideal I of C(X) exists a subset S of idempotents such that Ann(I)=Ann(S)? Along the line of answering these questions we introduce two classes of topological spaces. We call X an EF (resp., EZ)-space if disjoint unions of clopen sets are completely separated (resp., every regular closed subset is the closure of a union of clopen subsets). Topological properties of EF (resp., EZ)-spaces are investigated. As a consequence, a completely regular Hausdorff space X is an Fα-space in the sense of Comfort and Negrepontis for each infinite cardinal α if and only if X is an EF and EZ-space. Among other things, for a reduced ring R (resp., J(R)=0) we show that Spec(R) (resp., Max(R)) is an EZ-space if and only if for every ideal I of R there exists a subset S of idempotents of R such that Ann(I)=Ann(S).

Principally Quasi-Baer skew Generalized Power Series modules

Let R be a ring and S a strictly totally ordered monoid. Let ω: S → End(R) be a monoid homomorphism. Let M R be an ω-compatible module and either R satisfies the ascending chain conditions (ACC) on left annihilator ideals or every S-indexed subset of right semicentral idempotents in R has a generalized S-indexed join. We show that M R is p.q.-Baer if and only if the generalized power series module M[[S]] R[[S, ω]] is p.q.-Baer. As a consequence, we deduce that for an ω-compatible ring R, the skew generalized power series ring R[[S, ω]] is right p.q.-Baer if and only if R is right p.q.-Baer and either R satisfies the ACC on left annihilator ideals or any S-indexed subset of right semicentral idempotents in R has a generalized S-indexed join in R. Examples to illustrate and delimit the theory are provided.

Joseph Ideals and Harmonic Analysis for

The Joseph ideal in the universal enveloping algebra U(so(m)) is the annihilator ideal of the so(m)-representation on the harmonic functions on R^{m-2}. The Joseph ideal for sp(2n) is the annihilator ideal of the Segal-Shale-Weil (metaplectic) representation. Both ideals can be constructed in a unified way from a quadratic relation in the tensor algebra of g for g equal to so(m) or sp(2n). In this paper we construct two analogous ideals in the tensor algebra of g and U(g) for g the orthosymplectic Lie superalgebra osp(m|2n)=spo(2n|m) and prove that they have unique characterizations that naturally extend the classical case. Then we show that these two ideals are the annihilator ideals of respectively the osp(m|2n)-representation on the spherical harmonics on R^{m-2|2n} and a generalization of the metaplectic representation to spo(2n|m). This proves that these ideals are reasonable candidates to establish the theory of Joseph-like ideals for Lie superalgebras. We also discuss the relation between the Joseph ideal of osp(m|2n) and the algebra of symmetries of the super conformal Laplace operator, regarded as an intertwining operator between principal series representations for osp(m|2n).

Principally Quasi-Baer Skew Power Series Modules

A module M R is called principally quasi-Baer (or simply p.q.-Baer) if the annihilator of every cyclic submodule of M R is generated by an idempotent, as a right ideal. Let α be an automorphism of R and M R be an α-compatible module and every countable subset of right semicentral idempotents in R has a generalized countable join or R satisfies the ACC on left annihilator ideals. It is shown that M R is p.q.-Baer if and only if M[[x]] R[[x; α]] is p.q.-Baer if and only if M[[x, x −1]] R[[x, x −1; α]] is p.q.-Baer. As a consequence, we unify and extend nontrivially many of the previously known results such as [11, 15, 20]. Examples to illustrate and delimit the theory are provided.

ON ANNIHILATOR IDEALS OF C-ALGEBRAS

The concept of annihilator ideals is introduced in C-algebras. Some properties of these annihilator ideals are studied and then proved that the class of all annihilator ideals forms a complete Boolean algebra. A set of equivalent conditions are obtained for every ideal of a C-algebra to become an annihilator ideal. Some properties of homomorphic images and inverse images of annihilators ideals of a C-algebra are studied.

On annihilator ideals in matrix near-rings

On annihilator ideals in matrix near-rings

On Weak Annihilator Ideals of Skew Monoid Rings

As generalizations of annihilators and associated primes, we introduce the notions of weak annihilators and weak associated primes, respectively. We first study the properties of the weak annihilator of a subset X in a ring R. We next investigate how the weak associated primes of a ring R behave under passage to the skew monoid ring R*M. Let R be a semicommutative ring, and M an ordered monoid and φ: M → Aut(R) a compatible monoid homomorphism. Then we can describe all weak associated primes of the skew monoid ring R*M in terms of the weak associated primes of R in a very straightforward way.

On a Hereditary Radical Property Relating to the Reducedness

In this note, a hereditary radical property, called homomorphically reduced rings, is introduced, observed, and applied. The dual concept of this property is also studied with the help of Courter, proving that any ring R (possibly without identity) has an ideal S such that S/K is not homomorphically reduced for each proper ideal K of S; and if L is an ideal of R with L ⊊ S, then L/H is homomorphically reduced for some ideal H of R with H ⊊ L. The concept of the homomorphical reducedness is shown to be equivalent to the left (right) weak regularity and the (strong) regularity for one-sided duo rings. It is proved that homomorphically reduced rings have several useful properties similar to those of (weakly) regular rings. It is proved that the homomorphical reducedness can go up to classical quotient rings. It is shown that if R is a reduced right Ore ring with the ascending chain condition (ACC) for annihilator ideals, then the maximal right quotient ring of R is strongly regular (hence homomorphically reduced).

Semiprime elements and relative annihilators in multiplicative lattices

As generalization of prime element, semi prime element in a multiplicative lattice L is defined. Several characterizations of semi prime elements in terms of special relative annihilator ideals are obtained.

On Bochner-Martinelli residue currents and their annihilator ideals

Nous etudions le courant residuel de type Bochner-Martinelli R f associe a un m-uple de germes de fonctions holomorphes f = (f 1 ,..., f m ) definies a l'origine 0 ∈ C n dont l'ensemble des zeros communs se reduit a 0. Nos resultats principaux sont: une description geometrique de R f en terme des valuations de Rees associees a l'ideal (f) engendre par f et la caracterisation du cas ou l'ideal annihilateur de R f est egal a (f).

Residue currents with prescribed annihilator ideals

Given a coherent ideal sheaf J we construct locally a vector-valued residue current R whose annihilator is precisely the given sheaf. In case J is a complete intersection, R is just the classical Coleff–Herrera product. By means of these currents we can extend various results, previously known for a complete intersection, to general ideal sheaves. Combining with integral formulas we obtain a residue version of the Ehrenpreis–Palamodov fundamental principle.

ANNIHILATORS OF PRINCIPAL IDEALS IN THE EXTERIOR ALGEBRA

In this paper we describe annihilators of principal ideals of exterior algebras. For odd elements we establish formulae for dimensions of their principal ideals and their annihilators. For even elements we exhibit (multiplicative) generators for annihilator ideals.

Annihilators of principal ideals in the exterior algebra

In this paper we describe annihilators of principal ideals of exterior algebras. For odd elements we establish formulae for dimensions of their principal ideals and their annihilators. For even elements we exhibit (multiplicative) generators for annihilator ideals.

Central Sets and Radii of the Zero-Divisor Graphs of Commutative Rings

For a commutative ring R with identity, the zero-divisor graph, Γ(R), is the graph with vertices the nonzero zero-divisors of R and edges between distinct vertices x and y whenever xy = 0. This article gives a proof that the radius of Γ(R) is 0, 1, or 2 if R is Noetherian. The center union {0} is shown to be a union of annihilator ideals if R is Artinian. The diameter of Γ(R) can be determined once the center is identified. If R is finite, then the median is shown to be a subset of the center. A dominating set of Γ(R) is constructed using elements of the center when R is Artinian. It is shown that for a finite ring R ≇ ℤ2 × F for some finite field F, the domination number of Γ(R) is equal to the number of distinct maximal ideals of R. Other results on the structure of Γ(R) are also presented.

Power Series Rings Satisfying a Zero Divisor Property

In this note we continue to study zero divisors in power series rings and polynomial rings over general noncommutative rings. We first construct Armendariz rings which are not power-serieswise Armendariz, and find various properties of (power-serieswise) Armendariz rings. We show that for a semiprime power-serieswise Armendariz (so reduced) ring R with a.c.c. on annihilator ideals, R[[x]] (the power series ring with an indeterminate x over R) has finitely many minimal prime ideals, say B 1,…,B m , such that B 1… B m = 0 and B i = A i [[x]] for some minimal prime ideal A i of R for all i, where A 1,…,A m are all minimal prime ideals of R. We also prove that the power-serieswise Armendarizness is preserved by the polynomial ring extension as the Armendarizness, and construct various types of (power-serieswise) Armendariz rings.

Decomposition of tableaus annihilated by zero-dimensional ideals

A (1-dimensional) linear recursive sequence s over a field K decomposes in the following canonical way. Let I⊂ K[ x] be the annihilator ideal of s. Since K[ x] is a principal ideal domain, I= fK[ x] for some polynomial f, which can be factored as f 1⋯ f r where the f i are coprime. Thus s can be uniquely written as a sum of sequences s i having annihilator ideals I i = f i K[ x]. Furthermore, each f i is a power of an irreducible polynomial f i =( g i ) e i . Each sequence s i can be uniquely written as a e i -fold sum of pointwise products of a “binomial” sequence with a sequence annihilated by g i . Finally, a sequence annihilated by an irreducible polynomial g i is given by a trace formula. See, for instance, [N. Zierler, W.H. Mills, J. Algebra 27 (1973) 147–157]. We show that a completely analogous decomposition (which subsumes the 1-dimensional case) holds for n-dimensional linear recursive sequences, i.e., tableaus annihilated by zero-dimensional ideals of K[ x 1,…, x n ].

Injective modules over Mori domains

Injective modules are considered over commutative domains. It is shown that every injective module admits a decomposition into two summands, where one of the summands contains an essential submodule whose elements have divisorial annihilator ideals, while the other summand contains no element with divisorial annihilator. In the special case of Mori domains (i.e., the divisorial ideals satisfy the maximum condition), the first summand is a direct sum of a S-injective module and a module that has no such summand. The former is a direct sum of indecomposable injectives, while the latter is the injective hull of such a direct sum. Those Mori domains R are characterized for which the injective hull of Q/R is S-injective (Q denotes the field of quotients of R) as strong Mori domains, correcting a false claim in the literature.