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Salmanpour M.R. ,
Alizadeh M. ,
Mousavi G. ,
Sadeghi S. ,
Amiri S. ,
Oveisi M. ,
Rahmim a., ,
Hacihaliloglu I. ,
Moradzadehdehkordi, A. ,
Faghihi a., ,
Alagöz, Y. ,
Mohammadi hassanabadi a., A.M. IEEE Access (21693536) pp. 47217-47229
This study evaluates metrics for tasks such as classification, regression, clustering, correlation analysis, statistical tests, segmentation, and image-to-image (I2I) translation in medical imaging domains. Metrics were compared across Python libraries, R packages, and Matlab functions to assess their consistency and highlight discrepancies. The findings underscore the need for a unified roadmap to standardize metrics, ensuring reliable and reproducible ML evaluations across platforms. This study examined a wide range of evaluation metrics across various tasks in medical imaging and found only some to be consistent across platforms, such as Accuracy, Balanced Accuracy, Cohens Kappa, F-beta Score, MCC, Geometric Mean, AUC, and Log Loss in binary classification; Accuracy, Cohens Kappa, and F-beta Score in multi-class classification; MAE, MSE, RMSE, MAPE, Explained Variance, Median AE, MSLE, and Huber in regression; Davies-Bouldin Index and Calinski-Harabasz Index in clustering; Pearson, Spearman, Kendall’s Tau, Mutual Information, Distance Correlation, Bicor, Percbend, Shepherd, and Partial Correlation in correlation analysis; Paired t-test, Chi-Square Test, ANOVA, Kruskal-Wallis Test, Shapiro-Wilk Test, Welch’s t-test, and Bartlett’s test in statistical tests; Accuracy, Precision, and Recall in 2D segmentation; Accuracy in 3D segmentation; MAE, MSE, RMSE, and R-Squared in 2D-I2I translation; and MAE, MSE, and RMSE in 3D-I2I translation. Given observation of discrepancies in a number of metrics (e.g. precision, recall and F1 score in binary classification, WCSS in clustering, and multiple statistical tests, amongst multiple metrics), this study concludes that ML evaluation metrics require standardization and recommends that future research use consistent metrics for different tasks to effectively compare ML techniques and solutions.INDEX TERMS 2D/3D medical images, consistency of evaluation metrics in multi-framework, evaluation metric roadmap, ML evaluation metrics. © 2013 IEEE.
Journal of Algebra and its Applications (17936829)
In this paper, we continue to study weakly uniserial modules and rings. A module is called weakly uniserial if its submodules are comparable regarding embedding. Also, a right weakly uniserial ring is a ring which is weakly uniserial as a right module over itself. In addition to providing more properties of these modules, we introduce and investigate weakly serial modules (which are a direct sum of weakly uniserial modules). A right weakly serial ring is a ring which is weakly serial as a right module over itself. It is shown that over a right Artinian local ring R, every (2-generated) right R-module with a semiprime annihilator is weakly uniserial. The converse is true when R is a commutative Noetherian ring. We prove that an injective Z-module is weakly uniserial if and only if it is isomorphic to Zp∞, for some prime number p. In a weakly serial module we show that every its nonzero submodule contains a weakly uniserial submodule and every its fully invariant submodule is weakly serial. Also, it is shown that a right duo right self-injective ring is right weakly serial if and only if every projective right module is (weakly) serial. © 2026 World Scientific Publishing Company.
Hacettepe Journal of Mathematics and Statistics (2651477X) 53(2)pp. 342-355
In a recent paper, Mao has studied min-pure injective modules to investigate the existence of min-injective covers. A min-pure injective module is one that is injective relative only to min-pure exact sequences. In this paper, we study the notion of min-pure projective modules which is the projective objects of min-pure exact sequences. Various ring characterizations and examples of both classes of modules are obtained. Along this way, we give conditions which guarantee that each min-pure projective module is either injective or projective. Also, the rings whose injective objects are min-pure projective are consid-ered. The commutative rings over which all injective modules are min-pure projective are exactly quasi-Frobenius. Finally, we are interested with the rings all of its modules are min-pure projective. We obtain that a ring R is two-sided Köthe if all right R-modules are min-pure projective. Also, a commutative ring over which all modules are min-pure projective is quasi-Frobenius serial. As consequence, over a commutative indecomposable ring with J(R)2 = 0, it is proven that all R-modules are min-pure projective if and only if R is either a field or a quasi-Frobenius ring of composition length 2. © 2024, Hacettepe University. All rights reserved.
Rendiconti del Circolo Matematico di Palermo (0009725X) 73(8)pp. 3175-3193
Given modules M and A, M is said to be A-RD-subinjective if for every RD-extension B of A, every f∈Hom(A,M) extends to Hom(B,M). For a module M, the RD-subinjectivity domain of M is defined to be the collection of all modules A such that M is A-RD-subinjective. We investigate basic properties of RD-subinjectivity domains and provide characterizations for various types of rings and modules including p-injective modules, RD-coflat modules, von Neumann regular rings, RD-rings, Köthe rings, right Noetherian rings, and quasi-Frobenius rings in terms of RD-subinjectivity domains. Finally, we study the properties of RD-indigent modules and consider the structure of rings over which every (resp. simple) right module is RD-injective or RD-indigent. © The Author(s), under exclusive licence to Springer-Verlag Italia S.r.l., part of Springer Nature 2024.
Communications in Algebra (00927872) 51(11)pp. 4821-4829
Recall that an R-module M is pure-semisimple if every module in the category (Formula presented.) is a direct sum of finitely generated (and indecomposable) modules. A theorem from commutative algebra due to Köthe, Cohen-Kaplansky and Griffith states that “a commutative ring R is pure-semisimple (i.e., every R-module is a direct sum of finitely generated modules) if and only if every R-module is a direct sum of cyclic modules, if and only if, R is an Artinian principal ideal ring”. Consequently, every (or finitely generated, cyclic) ideal of R is pure-semisimple if and only if R is an Artinian principal ideal ring. Therefore, a natural question of this sort is “whether the same is true if one only assumes that every proper ideal of R is pure-semisimple?” The goal of this paper is to answer this question. The structure of such rings is completely described as Artinian principal ideal rings or local rings R with the maximal ideals (Formula presented.) which Rx is Artinian uniserial and T is semisimple. Also, we give several characterizations for commutative rings whose proper principal (finitely generated) ideals are pure-semisimple. © 2023 Taylor & Francis Group, LLC.
Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales - Serie A: Matematicas (15791505) 116(4)
A theorem due to Warfield states that “a ring R is left serial if and only if every (finitely generated) projective left R-module is serial” and a theorem due to Tuganbaev states that “a ring R is a finite direct product of uniserial Noetherian rings if and only if R is left duo, and all injective left R-modules are serial”. Most recently, in our previous paper [Virtually uniserial modules and rings, J Algebra 549:365–385, 2020], we introduced and studied the concept of virtually uniserial modules as a nontrivial generalization of uniserial modules. We say that an R-module M is virtually uniserial if, for every finitely generated submodule 0 ≠ K⊆ M, K/ Rad (K) is virtually simple (an R-module M is virtually simple if, M≠ 0 and M≅ N for every nonzero submodule N of M). Also, an R-module M is called virtually serial if it is a direct sum of virtually uniserial modules. The above results of Warfield and Tuganbaev motivated us to study the following questions: “Which rings have the property that every projective module is virtually serial?” and “Which rings have the property that every injective module is virtually serial?”. The goal of this paper is to answer these questions. © 2022, The Author(s) under exclusive licence to The Royal Academy of Sciences, Madrid.
Journal of Pure and Applied Algebra (00224049) 226(9)
We say that a ring R is a right RDV-ring if each simple right R-module is RD-injective. In this paper, we study the notion of RDV-rings which is a non-trivial generalization of V-rings and Köthe rings. For instance, commutative RD-rings, serial rings and right duo right uniserial rings are RDV-rings. Several characterizations of right RDV-rings are given. Also, it is shown that over a semilocal ring R with Jacobson radical J, each simple right R-module is RD-flat if and only if R is a left RDV-ring, if and only if R(R/J) is RD-injective, if and only if (R/J)R is RD-flat. As a consequence, we show that a local ring R is a principal ideal ring if and only if R satisfies the ascending chain condition on principal left ideals and R(R/J) is RD-injective. In the case of R being either a local left perfect ring or a normal left perfect ring, we have obtained results which state that to check whether every left R-module is RD-injective (or, R is Köthe), it suffices to test only the RD-injectivity of the simple left R-modules. Finally, we give some characterizations of quasi-Frobenius rings by using these concepts. © 2022 Elsevier B.V.
Communications in Algebra (00927872) 49pp. 3837-3849
We study the class of virtually homo-uniserial modules and rings as a nontrivial generalization of homo-uniserial modules and rings. An R-module M is virtually homo-uniserial if, for any finitely generated submodules (Formula presented.) the factor modules (Formula presented.) and (Formula presented.) are virtually simple and isomorphic (an R-module M is virtually simple if, (Formula presented.) and (Formula presented.) for every nonzero submodule N of M). Also, an R-module M is called virtually homo-serial if it is a direct sum of virtually homo-uniserial modules. We obtain that every left R-module is virtually homo-serial if and only if R is an Artinian principal ideal ring. Also, it is shown that over a commutative ring R, every finitely generated R-module is virtually homo-serial if and only if R is a finite direct product of almost maximal uniserial rings and principal ideal domains with zero Jacobson radical. Finally, we obtain some structure theorems for commutative (Noetherian) rings whose every proper ideal is virtually (homo-)serial. © 2021 Taylor & Francis Group, LLC.
Bulletin of the Korean Mathematical Society (10158634) 57(2)pp. 371-381
A ring R is called right pure-injective if it is injective with respect to pure exact sequences. According to a well known result of L. Melkersson, every commutative Artinian ring is pure-injective, but the converse is not true, even if R is a commutative Noetherian local ring. In this paper, a series of conditions under which right pure-injective rings are either right Artinian rings or quasi-Frobenius rings are given. Also, some of our results extend previously known results for quasi-Frobenius rings. ©2020 Korean Mathematial Soiety.
Journal of Algebra (00218693) 549pp. 365-385
We study the class of virtually uniserial modules and rings as a nontrivial generalization of uniserial modules and rings. An R-module M is virtually uniserial if for every finitely generated submodule 0≠K⊆M, K/Rad(K) is virtually simple. Also, an R-module M is called virtually serial if it is a direct sum of virtually uniserial modules and a left virtually uniserial (resp., left virtually serial) ring is a ring which is virtually uniserial (resp., serial) as a left R-module. We give some useful properties of virtually (uni)serial modules and rings. In particular, it is shown that every left virtually uniserial module is uniform and Bézout. Also, we show that if R is a left virtually serial ring, then R/J(R)≅∏i=1 tMn(Di) where t,n1,…,nt∈N and each Di is a principal left ideal domain. As a consequence, we obtain that a ring R is left virtually serial with J(R)=0 if and only if R≅∏i=1 tMn(Di) where t,n1,…,nt∈N and each Di is a principal left ideal domain with J(Di)=0. Also, several classes of rings for which every virtually uniserial module (resp., ring) is uniserial are given. Noetherian left virtually uniserial rings are characterized. Finally, we obtain some structure theorems for (commutative) rings over which every (finitely generated) module is virtually serial. © 2020 Elsevier Inc.
Journal of Algebra and its Applications (17936829) 18(1)
A left R-module M is said to be left singly injective if ExtR1(F/K,M) = 0 for any cyclic submodule K of any finitely generated free left R-module F. In this paper, we study the notion of singly injective modules which is generalization of injective modules and absolutely pure modules. In this direction, we give conditions which guarantee that each singly injective left R-module is either injective or absolutely pure. Finally, we study rings whose simple modules are singly injective (SSI-rings). © 2019 World Scientific Publishing Company.
Journal of Pure and Applied Algebra (00224049) 223(2)pp. 713-720
A famous theorem of commutative algebra due to I. M. Isaacs states that “if every prime ideal of R is principal, then every ideal of R is principal”. Therefore, a natural question of this sort is “whether the same is true if one weakens this condition and studies rings in which ideals are direct sums of cyclically presented modules?” The goal of this paper is to answer this question in the case R is a commutative local ring. We obtain an analogue of Isaacs's theorem. In fact, we give two criteria to check whether every ideal of a commutative local ring R is a direct sum of cyclically presented modules, it suffices to test only the prime ideals or structure of the maximal ideal of R. As a consequence, we obtain: if R is a commutative local ring such that every prime ideal of R is a direct sum of cyclically presented R-modules, then R is a Noetherian ring. Finally, we describe the ideal structure of commutative local rings in which every ideal of R is a direct sum of cyclically presented R-modules. © 2018 Elsevier B.V.
Journal of Algebra (00218693) 486pp. 422-424
The statement of Theorem 2.1 in [4] is not correct. We give a counterexample to this statement and following to this error we give further corrections. We also offer two better substitutions for Proposition 2.9 and Theorem 2.11 in [4]. © 2017 Elsevier Inc.
Journal of Pure and Applied Algebra (00224049) 221(4)pp. 935-947
We study direct-sum decompositions of pure-projective modules over some classes of rings. In particular, we determine several classes of rings over which every pure-projective left module is a direct sum of cyclic modules. Finally, the relationship between our results and FGC rings is also studied. © 2016 Elsevier B.V.
Journal of Algebra (00218693) 478pp. 419-436
A ring R is said to be left pure-hereditary (resp. RD-hereditary) if every left ideal of R is pure-projective (resp. RD-projective). In this paper, some properties and examples of these rings, which are nontrivial generalizations of hereditary rings, are given. For instance, we show that if R is a left RD-hereditary left nonsingular ring, then R is left Noetherian if and only if u.dim(RR)<∞. Also, we show that a ring R is quasi-Frobenius if and only if R is a left FGF, left coherent right pure-injective ring. A ring R is said to be left FP-hereditary if every left ideal of R is FP-projective. It is shown that if R is a left CF ring, then R is left Noetherian if and only if R is left pure-hereditary, if and only if R is left FP-hereditary, if and only if R is left coherent. It is shown that every left self-injective left FP-hereditary ring is semiperfect. Finally, it is shown that a ring R is left FP-hereditary (resp. left coherent) if and only if every submodule (resp. finitely generated submodule) of a projective left R-module is FP-projective, if and only if every pure factor module of an injective left R-module is injective (resp. FP-injective), if and only if for each FP-injective left R-module U, E(U)/U is injective (resp. FP-injective). © 2017 Elsevier Inc.
Colloquium Mathematicum (00101354) 145(2)pp. 167-177
A classical question due to Yoneda is, “When is the tensor product of any two injective modules injective?” Enochs and Jenda gave a complete and explicit answer to this question in 1991. Since RD-injective modules are a generalization of injective modules, it is natural to ask whether the tensor product of any two RD-injective modules is RD-injective. In this paper we deal with this question. © Instytut Matematyczny PAN, 2016.
Journal of Algebra (00218693) 460pp. 128-142
A famous theorem of algebra due to Osofsky states that "if every cyclic left R-module is injective, then R is semisimple". Therefore, a natural question of this sort is: "What is the class of rings R for which every cyclic left R-module is pure-injective or pure-projective?" The goal of this paper is to answer this question. For instance, we show that if every cyclic left R-module is pure-injective, then R is a left perfect ring. As a consequence, a commutative coherent ring R is Artinian if and only if every cyclic R-module is pure-injective. Also, a commutative ring R is pure-semisimple (i.e., every R-module is pure-injective) if and only if all cyclic R-modules and all indecomposable R-modules are pure-injective. We obtain some generalizations of Osofsky's theorem in the cases R is semiprimitive or commutative coherent or a commutative semiprime Goldie ring. Finally, we show that a ring R is left Noetherian if and only if every cyclic left R-module is pure-projective. As a corollary of this result we obtain: if every cyclic left R-module is pure-injective and pure-projective, then R is a left Artinian ring. The converse is also true when R is commutative. © 2016 Elsevier Inc.
Communications in Algebra (00927872) 42(5)pp. 2061-2081
In this article, several characterizations of certain classes of rings via FC-purity and I-purity are considered. Among others results, it is shown that every I-pure injective left R-module is projective if and only if every FC-pure projective left R-module is injective, if and only if, R is a semisimple ring. In particular, the structures of FC-pure projective and I-pure projective modules over a left Artinian ring are completely described. Also, it is shown that every left R-module is FC-pure projective if and only if every indecomposable left R-module is a finitely presented cyclic R-module, if and only if, R is a left Köthe ring. Finally, we introduce FC-pure flatness and I-pure flatness of modules and several characterizations of these notions are given. In particular, we show that a commutative ring R is quasi-Frobenius if and only if R is an Artinian ring and I-pure injective, if and only if, R is an Artinian ring and the injective envelope E(R) is an FC-pure projective R-module. © 2014 Copyright Taylor and Francis Group, LLC.
Proceedings of the American Mathematical Society (10886826) 142(8)pp. 2625-2631
In this paper, we obtain a partial solution to the following question of Köthe: For which rings R is it true that every left (or both left and right) R-module is a direct sum of cyclic modules? Let R be a ring in which all idempotents are central. We prove that if R is a left Köthe ring (i.e., every left R-module is a direct sum of cyclic modules), then R is an Artinian principal right ideal ring. Consequently, R is a Köthe ring (i.e., each left and each right R-module is a direct sum of cyclic modules) if and only if R is an Artinian principal ideal ring. This is a generalization of a Köthe-Cohen-Kaplansky theorem. © 2014, American Mathematical Society.
Journal of Algebra (00218693) 401pp. 179-200
We study direct-sum decompositions of RD-projective modules. In particular, we investigate the rings over which every RD-projective right module is a direct sum of cyclically presented right modules, or a direct sum of finitely presented cyclic right modules, or a direct sum of right modules with local endomorphism rings (SSP rings). SSP rings are necessarily semiperfect. For instance, the superlocal rings introduced by Puninski, Prest and Rothmaler in [28] and the semilocal strongly π-regular rings introduced by Kaplansky in [24] are SSP rings. In the case of a Noetherian ring R (with further additional hypotheses), an RD-projective R-module M turns out to be either a direct sum of finitely presented cyclic modules or of the form M = T(M) ⊕ P, where T(M) is the torsion part of M (elements of M annihilated by a regular element of R) and P is a projective module. © 2013 Elsevier Inc.
Communications in Algebra (00927872) 41(12)pp. 4559-4575
This paper investigates the structure of cyclically pure (or C-pure) projective modules. In particular, it is shown that a ring R is left Noetherian if and only if every C-pure projective left R-module is pure projective. Also, over a left hereditary Noetherian ring R, a left R-module M is C-pure projective if and only if M = N ⊕ P, where N is a direct sum of cyclic modules and P is a projective left R-module. The relationship Cpurity with purity and RD-purity are also studied. It is shown that if R is a local duoring, then the C-pure projective left R-modules and the pure projective left R-modules coincide if and only if R is a principal ideal ring. If R is a left perfect duo-ring, then the C-pure projective left R-modules and the pure projective left R-modules coincide if and only if R is left Köthe (i.e., every left R-module is a direct sum of cyclic modules). Also, it is shown that for a ring R, if every C-pure projective left R-module is RD-projective, then R is left Noetherian, every p-injective left R-module is injective and every p-flat right R-module is flat. Finally, it is shown that if R is a left p.p-ring and every C-pure projective left R-module is RD-projective, then R is left Noetherian hereditary. The converse is also true when R is commutative, but it does not hold in the noncommutative case. © Taylor & Francis Group, LLC.
Archivum Mathematicum (00448753) 48(4)pp. 291-299
In this paper we study commutative rings R whose prime ideals are direct sums of cyclic modules. In the case R is a finite direct product of commutative local rings, the structure of such rings is completely described. In particular, it is shown that for a local ring (R, Ai), the following statements are equivalent: (1) Every prime ideal of R is a direct sum of cyclic R-modules; (2) M =⊕λχλ Rωλ where λ is an index set and R/Ann(ωλ) is a principal ideal ring for each λ χ λ; (3) Every prime ideal of H is a direct sum of at most |A| cyclic fi-modules where A is an index set and M =⊕λχλ Rωλ; and (4) Every prime ideal of R is a summand of a direct sum of cyclic R-modules. Also, we establish a theorem which state that, to check whether every prime ideal in a Noetherian local ring (R, M) is a direct sum of (at most n) principal ideals, it suffices to test only the maximal ideal M.
Journal of Algebra (00218693) 345(1)pp. 257-265
A theorem from commutative algebra due to Köthe and Cohen-Kaplansky states that, "a commutative ring R has the property that every R-module is a direct sum of cyclic modules if and only if R is an Artinian principal ideal ring". Therefore, an interesting natural question of this sort is "whether the same is true if one only assumes that every ideal is a direct sum of cyclic modules?" The goal of this paper is to answer this question in the case R is a finite direct product of commutative Noetherian local rings. The structure of such rings is completely described. In particular, this yields characterizations of all commutative Artinian rings with this property. © 2011.
Bulletin of the Korean Mathematical Society (10158634) (4)pp. 663-673
This work was intended as an attempt to introduce and investigate the Connes-amenability of module extension of dual Banach algebras. It is natural to try to study the weak*-continuous derivations on the module extension of dual Banach algebras and also the weak Connesamenability of such Banach algebras. © 2010 The Korean Mathematical Society.
