Essential Socle Research Articles

Banach algebra mappings preserving the invertibility of linear pencils

Let A and B be complex unital Banach algebras, and let φ,ψ:A→B be surjective mappings. If A is semisimple with an essential socle and φ and ψ together preserve the invertibility of linear pencils in both directions, that is, for any x,y∈A and λ∈C, λx+y is invertible in A if and only if λφ(x)+ψ(y) is invertible in B, then we show that there exists an invertible element u in B and a Jordan isomorphism J:A→B such that φ(x)=ψ(x)=uJ(x) for all x∈A.

Essential ideals represented by mod-annihilators of modules

Let R be a commutative ring with unity, M be a unitary R-module and G a finite abelian group (viewed as a \({\mathbb {Z}}\)-module). The main objective of this paper is to study properties of mod-annihilators of M. For \(x \in M\), we study the ideals \([x : M] =\{r\in R ~|~ rM\subseteq Rx\}\) of R corresponding to mod-annihilator of M. We investigate as when [x : M] is an essential ideal of R. We prove that the arbitrary intersection of essential ideals represented by mod-annihilators is an essential ideal. We observe that [x : M] is injective if and only if R is non-singular and the radical of R/[x : M] is zero. Moreover, if essential socle of M is non-zero, then we show that [x : M] is the intersection of maximal ideals and \([x : M]^2 = [x : M]\). Finally, we discuss the correspondence of essential ideals of R and vertices of the annihilating graphs realized by M over R.

Spectrally additive group homomorphisms on Banach algebras

Let A and B be unital complex Banach algebras. A surjective map ϕ:A→B is called a spectrally additive group homomorphism if the spectrum of x±y is equal to the spectrum of ϕ(x)±ϕ(y) for each x,y∈A. If A is semisimple and either A or B has an essential socle, then we prove that a spectrally additive group homomorphism ϕ:A→B is a continuous Jordan-isomorphism. If, in addition, either A or B is prime, then we conclude that ϕ is either a continuous algebra isomorphism or anti-isomorphism. It is noteworthy that the continuity and linearity (or even additivity) of the map ϕ does not form part of the hypothesis, but is rather obtained in the conclusion. The techniques employed in the proof of these results utilizes the spectral rank, trace and determinant, and yields a new additive characterization of finite rank elements in a Banach algebra which is of independent interest.

Socle-injective semiprime rings, with some applications to Leavitt path algebras

A ring R is called right (or left) socle-injective if every R-homomorphism from the right (or left) socle of R into R extends to R. In this article, we show that any semiprime ring R with socle S, is socle-injective if and only if where Q′ is a suitable subring of maximal right ring of quotients of R and is an ideal of Q′. Furthermore, an explicit structure of the ring Q′ is presented for a semiprime socle- injective ring, with essential socle. As an application, we show that a unital Leavitt path algebra with essential socle is socle-injective if and only if is semisimple, hence von Neumann regular. Moreover, we observed that socle-injective Leavitt path algebras are left-right symmetric. We also have provided examples to illustrate our results.

When is every module with essential socle a direct sum of automorphism-invariant modules?

The purpose of this paper is to study the structure of rings over which every essential extension of a direct sum of a family of simple modules is a direct sum of automorphism-invariant modules. We show that if [Formula: see text] is a right quotient finite dimensional (q.f.d.) ring satisfying this property, then [Formula: see text] is right Noetherian. Also, we show a von Neumann regular (semiregular) ring [Formula: see text] with this property is Noetherian. Moreover, we prove that a commutative ring with this property is an Artinian principal ideal ring.

A note on peripherally multiplicative maps on Banach algebras

Let A and B be complex Banach algebras, and let ϕ,ϕ1, and ϕ2 be surjective maps from A onto B. Denote by ∂σ(x) the boundary of the spectrum of x. If A is semisimple, B has an essential socle, and ∂σ(xy)=∂σ(ϕ1(x)ϕ2(y)) for each x,y∈A, then we prove that the maps x↦ϕ1(1)ϕ2(x) and x↦ϕ1(x)ϕ2(1) coincide and are continuous Jordan isomorphisms. Moreover, if A is prime with nonzero socle and ϕ1 and ϕ2 satisfy the aforementioned condition, then we show once again that the maps x↦ϕ1(1)ϕ2(x) and x↦ϕ1(x)ϕ2(1) coincide and are continuous. However, in this case we conclude that the maps are either isomorphisms or anti-isomorphisms. Finally, if A is prime with nonzero socle and ϕ is a peripherally multiplicative map, then we prove that ϕ is continuous and either ϕ or −ϕ is an isomorphism or an anti-isomorphism.

The Socle of the Last Term in the Minimal Injective Resolution of a Gorenstein Module

Let $R$ be a left Noetherian ring, $S$ a right Noetherian ring and $_RU$ a Gorenstein module with $S={\rm End}(_RU)$. If the injective dimensions of $_RU$ and $U_S$ are finite, then the last term in the minimal injective resolution of $_{R}U$ has an essential socle.

On infinite direct sums of lifting modules

The aim of the present article is to investigate the structure of rings [Formula: see text] satisfying the condition: for any family [Formula: see text] of simple right [Formula: see text]-modules, every essential extension of [Formula: see text] is a direct sum of lifting modules, where [Formula: see text] denotes the injective hull. We show that every essential extension of [Formula: see text] is a direct sum of lifting modules if and only if [Formula: see text] is right Noetherian and [Formula: see text] is hollow. Assume that [Formula: see text] is an injective right [Formula: see text]-module with essential socle. We also prove that if every essential extension of [Formula: see text] is a direct sum of lifting modules, then [Formula: see text] is [Formula: see text]-injective. As a consequence of this observation, we show that [Formula: see text] is a right V-ring and every essential extension of [Formula: see text] is a direct sum of lifting modules for all simple modules [Formula: see text] if and only if [Formula: see text] is a right [Formula: see text]-V-ring.

Minus partial order and linear preservers

In this paper, we investigate the properties of minus partial order in unital rings. We generalize several results well known for matrices and bounded linear operators on Banach spaces. We also study linear maps preserving the minus partial order in unital semisimple Banach algebras with essential socle.

MODULES WHOSE CLOSED SUBMODULES WITH ESSENTIAL SOCLE ARE DIRECT SUMMANDS

We introduce and study CLESS-modules, which subsume two generalizations of extending modules due to P.F. Smith and A. Tercan. A module $M$ will be called a CLESS-module if every closed submodule $N$ of $M$ (in the sense that $M/N$ is non-singular) with essential socle is a direct summand of $M$. Various properties concerning direct sums of CLESS-modules are established. We show that, over a Dedekind domain, a module is CLESS if and only if its torsion submodule is a direct summand. We also study the behaviour of CLESS-modules under excellent extensions of rings.

Orthogonality preserving linear maps on [formula omitted]-algebras with non-zero socles

We study orthogonality preserving surjective linear maps from a unital C∗-algebra with non-zero socle A to a C∗-algebra. In particular, we prove that, when A has essential socle, every biorthogonality preserving surjective linear map T:A→B is automatically bounded and a Jordan ∗-isomorphism multiplied by an invertible element.

When Some Complement of an EC-Submodule is a Direct Summand

A module M is said to satisfy the condition if every ec-submodule of M has a complement which is a direct summand. We show that for a multiplication module over a commutative ring the and P-extending conditions are equivalent. It is shown that the property is not inherited by direct summands. Moreover, we prove that if M is an -module where SocM is an ec-submodule, then it is a direct sum of a module with essential socle and a module with zero socle. An example is given to show that the reverse of the last result does not hold.

Linear maps preserving the minimum modulus

We characterize surjective linear maps that preserve the minimum modulus between unital semisimple Banach algebras, one of them is a unital C ∗ -algebra having either real rank zero or essential socle. We also describe surjective linear maps on L(H), with H an infinite- dimensional Hilbert space, preserving the essential minimum modulus. Results concerning sur- jectivity and maximum modulus are also obtained.

Modules Whose Submodules are Essentially Embedded in Direct Summands

A module M is said to satisfy the C 12 condition if every submodule of M is essentially embedded in a direct summand of M. It is known that the C 11 (and hence also C 1) condition implies the C 12 condition. We show that the class of C 12-modules is closed under direct sums and also essential extensions whenever any module in the class is relative injective with respect to its essential extensions. We prove that if M is a -module with cancellable socle and satisfies ascending chain (respectively, descending chain) condition on essential submodules, then M is a direct sum of a semisimple and a Noetherian (respectively, Artinian) submodules. Moreover, a C 12-module with cancellable socle is shown to be a direct sum of a module with essential socle and a module with zero socle. An example is constructed to show that the reverse of the last result do not hold.

WHEN IS A SEMIPERFECT RING RIGHT PF?

It is well-known that a ring R is right PF if and only if it is semiperfect and right self-injective with essential right socle. In this note, it is shown that a ring R is right PF if and only if u.dim(RR) < ∞ and every colocal, injective right R-module is projective. Consequently, a semiperfect ring R is right PF if and only if the two classes of colocal injective right R-modules and local projective right modules coincide.

Inner ideal structure of nearly Artinian Lie algebras

In this paper we study the inner ideal structure of nondegenerate Lie algebras with essential socle, and characterize, in terms of the whole algebra, when the socle is Artinian.

A NOTE ON ESSENTIAL EXTENSIONS OF SEMI-SIMPLE MODULES

In a series of recent papers, Beidar, Jain and Srivastava studied the question as to when a ring R with the property that essential extensions of semi-simple right R-modules are direct sums of quasi-injectives is right Noetherian. Beidar and Jain proved that it is, when R is commutative or right q.f.d. In this note we extend their results proving the following: A ring R with this property is right Noetherian iff for some n ∈ ℕ, R/socn(RR) has ascending chain condition on essential non-two-sided right ideals (in particular, when R/socn(RR) is right q.f.d. or commutative). Also shown is the following: A ring is a right Noetherian right V-ring iff modules with essential socle are quasi-continuous/quasi-injective.

Algebras of quotients of path algebras

Leavitt path algebras are shown to be algebras of right quotients of their corresponding path algebras. Using this fact we obtain maximal algebras of right quotients from those (Leavitt) path algebras whose associated graph satisfies that every vertex connects to a line point (equivalently, the Leavitt path algebra has essential socle). We also introduce and characterize the algebraic counterpart of Toeplitz algebras.

When is Every Module with Essential Socle a Direct Sums of Quasi-Injectives?

ABSTRACT In the present article we study the structure of rings, over which essential extensions of semisimple modules are direct sums of quasi-injectives. In the special case of commutative rings, these rings are precisely Artinian PIR and so every module over such rings is a direct sum of cyclics as characterized by Köthe and Cohen-Kaplansky.

ADDENDUM AND CORRIGENDUM TO “WEAK (C11 +) MODULES WITH ACC OR DCC ON ESSENTIAL SUBMODULES”

It has come to our attention that there is a positive answer to the following question mentioned in [2, p. 734] : ``We may conjecture whether a $(WC_{11})$-module with essential socle is a $(C_{11})$-module? ". Unfortunately in [2, Example 9] we mistakenly stated that \begin{eqnarray*} R=\Big[\begin{array}{ll} \mathbb{Z}/\mathbb{Z}p^{2} & \mathbb{Z}/\mathbb{Z}p^{2} \\ 0 & \mathbb{Z}/\mathbb{Z}p^{2} \end{array}\Big] \end{eqnarray*} is not right $(C_{11})$. However this $R$ is, in fact, right $(C_{11})$ (see, [1, Theorem 2.4]). Moreover the following proposition provides the answer to the question which do not affect the other results in the paper.