Abstract
In this paper, we investigate certain class of submodules which contains that of superfluous submodules. A submodule W of an R-module M is annihilator large-superfluous, if ℓ
S
(V) ≠ 0 ...implies that W + V ≠ M where V is a large in M and S = End
R
(M). Several properties and characterizations of such submodules are consider. For α∈ S, we study under what conditions the image of α, Im(α) being annihilator large – superfluous submodule in M. We show that W
S
(M) = { α∈ S │Im(α) is annihilator large-superfluous in M } equal to { α∈ S │ lm(α) is large-superfluous } under certain class of projectivity. The sum E
R
(M) of all such submodules of M contains J
e
(M) and Z
s
(M). If M is cyclic, then E
R
(M) is the unique largest annihilator large-superfluous in M. MSC (2010): Primary: 16010; Secondary 16080.
Abstract
Through this paper R represent a commutative ring with identity and all R-modules are unitary left R-modules. In this work we consider a generalization of the class of essential submodules ...namely annihilator essential submodules. We study the relation between the submodule and his annihilator and we give some basic properties. Also we introduce the concept of annihilator uniform modules and annihilator maximal submodules.
Let R be a commutative ring with nonzero identity, Z(R) be its set of zero-divisors, and if a ∈ Z(R), then let ann
R
(a) = {d ∈ R | da = 0}. The annihilator graph of R is the (undirected) graph AG(R) ...with vertices Z(R)* = Z(R)∖{0}, and two distinct vertices x and y are adjacent if and only if ann
R
(xy) ≠ ann
R
(x) ∪ ann
R
(y). It follows that each edge (path) of the zero-divisor graph Γ(R) is an edge (path) of AG(R). In this article, we study the graph AG(R). For a commutative ring R, we show that AG(R) is connected with diameter at most two and with girth at most four provided that AG(R) has a cycle. Among other things, for a reduced commutative ring R, we show that the annihilator graph AG(R) is identical to the zero-divisor graph Γ(R) if and only if R has exactly two minimal prime ideals.
On torsion elements and their annihilators Abdollah, Zahra; Malakooti Rad, Parastoo; Ghalandarzadeh, Shaban ...
Journal of algebra,
11/2022, Volume:
610
Journal Article
Peer reviewed
Let R be a commutative ring with identity, and let M be an R-module. In this paper, we focus on the ideals of R that are annihilators of torsion elements of M. In analogy with definitions and results ...on zero-divisor and annihilator graphs of rings, we define the annihilator graph of a module. We investigate the structure, the diameter, and the girth of this graph and the closely related torsion graph of a module introduced by Ghalandarzadeh and Malakooti Rad. Given the right definitions, the properties of the modules are reflected in the graph theoretic properties of the graphs. We thus modify and extend results on zero divisors of rings to the much more general setting of modules and their torsion elements. In addition, we significantly strengthen the known results on the torsion graphs of modules. Some of the results will be refined further for the cases when M is a multiplication module or a reduced module.
Let R be a commutative ring with identity, and let Z(R) be the set of zero-divisors of R. The annihilator graph of R is defined as the undirected graph AG(R) with the vertex set Z(R)* = Z(R) \ ...{0}, and two distinct vertices x and y are adjacent if and only if ann_R(xy) \neq ann_R(x) \cup ann_R(y). In this paper, all rings whose annihilator graphs can be embedded on the plane or torus are classified.
•Four isomerized DPA-An dimers are used as annihilators of TTA-UC.•Connecting TTA efficiency and molecular structure from an energetic perspective.•Upconversion efficiency is inversely proportional ...to ΔE2T1-T3.•ΔE2T1-S1 > 0 is requisite but no need > 0.3 eV due to larger energy losses.
The enhancement in the efficiency of triplet-triplet annihilation upconversion (TTA-UC) is mainly determined by the triplet energy transfer (TET) and triplet-triplet annihilation (TTA) between the sensitizers and annihilators. The TET process works efficiently by adjusting the concentration ratio of the sensitizers and annihilators. The efficiency of TTA is determined by the properties of the annihilator. Because TTA is a Dexter-type energy transfer and is affected by the diffusion rate, the energy levels of the excited states and the molecular size are both crucial in TTA. In this study, four isomerized dimers of 9,10-diphenlanthracene (DPA) and anthracene (An) were designed and prepared as annihilators for TTA-UC. The singlet and triplet energy levels could be adjusted by altering the connection position while maintaining the molecular weight and size. When PtOEP was used as the sensitizer, the maximum upconversion efficiency of 9-4-(9-anthracenyl)phenyl-10-phenylanthracene (9DPA-9An) was ∼11.18%. This is four times higher than that of 9,10-diphenyl-2,9′-bianthracene (2DPA-9An, 2.63%). The calculation of the energies of T1 and the higher triplet state (T3, because E(T2) is similar to the E(T1) of these dimers) for these dimers has provided insights into the underlying reasons. These indicated that the energy gap value of 2 × E(T1) − E(T3) is the determining factor for TTA efficiency. This work may provide a better understanding of the excited-state energy levels, which is crucial for designing novel annihilators to enhance the TTA-UC efficiency.
When four isomerized DPA-An dimers as TTA-UC annihilators, 9DPA-9An can realize the relatively highest UC efficiency because the smallest ΔE2T1-T3. Display omitted
In this article, we introduce the weak Armendariz ideals as a generalization of the Armendariz ideals and we examine its properties, its relation to other structures. Also by giving numerous examples ...and diverse, we evaluate the behavior of weak Armendariz ideals under some ring extensions../files/site1/files/71/14.pdf
In this note, we give a counterexample and modification of Lemma 4.4 (ii) in
1
. Moreover, the validity of Theorem 4.6 in
1
is confirmed without using Lemma 4.4 in its proof.
As indispensable molecular components, photosensitizers play a crucial role in determining the quantum efficiency of triplet–triplet annihilation upconversion (TTA UC). This emergent technology has ...attracted great attention in recent years for realizing large anti‐Stokes shifts with noncoherent excitation sources. In a typical TTA UC, low‐energy photons are first harvested by the photosensitizers, which upon intersystem crossing (ISC) undergo triplet–triplet energy transfer (TTET) to emitters (i.e., annihilators). Following the bimolecular TTA among the emitters, high‐energy photons are given off by the singlet excited state of the emitters. Apparently, the efficiencies of photon absorption, ISC, and TTET are all dependent on the sensitizers. With a Dexter‐type ET mechanism requiring collisional interactions, a long triplet lifetime of the energy donor (photosensitizer) is evidently favorable for enhancing the efficiency of TTET. This progress report summarizes the recent developments of photosensitizers used for TTA UC, many of which feature a bichromophoric molecular scaffold. Among the various consequences and functions entailed by such bichromophoric designs, the extended triplet lifetime is a particularly advantageous property for TTA UC. Additionally, these new potent photosensitizers with long triplet lifetimes are also useful for other applications such as singlet oxygen sensitization and oxygen sensing.
Molecular design principles of photosensitizers applicable to triplet–triplet annihilation upconversion (TTA UC) are discussed. Bichromophoric molecular scaffolds have been widely adopted recently. In addition to enhancing light harvesting, the bichromophoric designs are effective at extending triplet lifetime of the system. Such a unique property ensures efficient energy transfer from sensitizer to annihilator and promotes efficient TTA UC.
PDF
