Almost Perfect Commutative Rings
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Arxiv:2005.02059V1 [Math.RA]
A new approach to Baer and dual Baer modules with some applications N. Ghaedan 1 and M.R. Vedadi 2 Department of Mathematical Sciences, Isfahan University of Technology, Isfahan, 84156-83111, IRAN. ABSTRACT. Let R be a ring. It is proved that an R-module M is Baer (resp. dual Baer) if and only if every exact sequence 0 → X → M → Y → 0 with Y ∈ Cog(MR) (resp. X ∈ Gen(MR)) splits. This shows that being (dual) Baer is a Morita invariant property. As more + applications, the Baer condition for the R-module M = HomZ(M, Q/Z) is investigated and shown that R is a von Neumann regular ring, if R+ is a Baer R-module. Baer modules with (weak) chain conditions are studied and determined when a Baer (resp. dual baer) module is a direct sum of mutually orthogonal prime (resp. co-prime) modules. While finitely generated dual Baer modules over commutative rings is shown to be semisimple, finitely generated Baer modules over commutative domain are studied. In particular, if R is commutative hereditary Noetherian domain then a finitely generated MR is Baer if and only if it is projective or semisim- ple. Over a right duo perfect ring, it is shown that every (dual) Baer modules is semisimple. Keywords: Baer module, character module, co-prime module, prime module, dual baer, regular ring, retractable module. MSC(2010): Primary: 16D10; 16D40 Secondary: 13C05; 13C10. 1. Introduction Throughout rings will have unit elements and modules will be right unitary. A ring R is said to be Baer if for every non-empty subset X of R, the right annihilator X in R is of the form eR for some e = e2 ∈ R. -
New Formulas for Semi-Primes. Testing, Counting and Identification
New Formulas for Semi-Primes. Testing, Counting and Identification of the nth and next Semi-Primes Issam Kaddouraa, Samih Abdul-Nabib, Khadija Al-Akhrassa aDepartment of Mathematics, school of arts and sciences bDepartment of computers and communications engineering, Lebanese International University, Beirut, Lebanon Abstract In this paper we give a new semiprimality test and we construct a new formula for π(2)(N), the function that counts the number of semiprimes not exceeding a given number N. We also present new formulas to identify the nth semiprime and the next semiprime to a given number. The new formulas are based on the knowledge of the primes less than or equal to the cube roots 3 of N : P , P ....P 3 √N. 1 2 π( √N) ≤ Keywords: prime, semiprime, nth semiprime, next semiprime 1. Introduction Securing data remains a concern for every individual and every organiza- tion on the globe. In telecommunication, cryptography is one of the studies that permits the secure transfer of information [1] over the Internet. Prime numbers have special properties that make them of fundamental importance in cryptography. The core of the Internet security is based on protocols, such arXiv:1608.05405v1 [math.NT] 17 Aug 2016 as SSL and TSL [2] released in 1994 and persist as the basis for securing dif- ferent aspects of today’s Internet [3]. The Rivest-Shamir-Adleman encryption method [4], released in 1978, uses asymmetric keys for exchanging data. A secret key Sk and a public key Pk are generated by the recipient with the following property: A message enciphered Email addresses: [email protected] (Issam Kaddoura), [email protected] (Samih Abdul-Nabi) 1 by Pk can only be deciphered by Sk and vice versa. -
Some New Results on Odd Perfect Numbers
Pacific Journal of Mathematics SOME NEW RESULTS ON ODD PERFECT NUMBERS G. G. DANDAPAT,JOHN L. HUNSUCKER AND CARL POMERANCE Vol. 57, No. 2 February 1975 PACIFIC JOURNAL OF MATHEMATICS Vol. 57, No. 2, 1975 SOME NEW RESULTS ON ODD PERFECT NUMBERS G. G. DANDAPAT, J. L. HUNSUCKER AND CARL POMERANCE If ra is a multiply perfect number (σ(m) = tm for some integer ί), we ask if there is a prime p with m = pan, (pa, n) = 1, σ(n) = pα, and σ(pa) = tn. We prove that the only multiply perfect numbers with this property are the even perfect numbers and 672. Hence we settle a problem raised by Suryanarayana who asked if odd perfect numbers neces- sarily had such a prime factor. The methods of the proof allow us also to say something about odd solutions to the equation σ(σ(n)) ~ 2n. 1* Introduction* In this paper we answer a question on odd perfect numbers posed by Suryanarayana [17]. It is known that if m is an odd perfect number, then m = pak2 where p is a prime, p Jf k, and p = a z= 1 (mod 4). Suryanarayana asked if it necessarily followed that (1) σ(k2) = pa , σ(pa) = 2k2 . Here, σ is the sum of the divisors function. We answer this question in the negative by showing that no odd perfect number satisfies (1). We actually consider a more general question. If m is multiply perfect (σ(m) = tm for some integer t), we say m has property S if there is a prime p with m = pan, (pa, n) = 1, and the equations (2) σ(n) = pa , σ(pa) = tn hold. -
TORSION THEORIES OVER SEMIPERFECT RINGS (A) ¿Rnjs
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 34, Number 2, August 1972 TORSION THEORIES OVER SEMIPERFECT RINGS EDGAR A. RUTTER, JR. Abstract. In this note we characterize those torsion theories over a semiperfect ring such that the class of torsion free modules is closed under projective covers and the hereditary torsion theories for which this is true of the class of torsion modules. These results are applied to determine all hereditary torsion theories over an injective cogenerator ring with the property that the torsion sub- module of every module is a direct summand. The purpose of this note is to investigate some properties of torsion theories in the category of modules over a semiperfect ring. We charac- terize those torsion theories with the property that the class of torsion free modules is closed under projective covers and the hereditary torsion theories for which this is true of the class of torsion modules. A torsion theory is called splitting provided the torsion submodule of every module is a direct summand. Bernhardt [4] proved that every splitting hereditary torsion theory for a quasi-Frobenius ring is determined by a central idempotent of the ring. The results mentioned above permit us to extend this result to a natural generalization of quasi-Frobenius rings, namely, the one-sided injective cogenerator rings. Bernhardt's theorem depended on a result of Alin and Armendariz [1] which states that every hereditary torsion class over a right perfect ring is closed under direct products and hence is determined by an idempotent ideal of the ring. While we shall not require this result our methods yield a new proof of it. -
S-ALMOST PERFECT COMMUTATIVE RINGS Contents Introduction 1 1. Preliminaries 3 2. S-Divisible and S-Torsion Modules 5 3. S-Strong
S-ALMOST PERFECT COMMUTATIVE RINGS SILVANA BAZZONI AND LEONID POSITSELSKI Abstract. Given a multiplicative subset S in a commutative ring R, we consider S-weakly cotorsion and S-strongly flat R-modules, and show that all R-modules have S-strongly flat covers if and only if all flat R-modules are S-strongly flat. These equivalent conditions hold if and only if the localization RS is a perfect ring and, for every element s 2 S, the quotient ring R=sR is a perfect ring, too. The multiplicative subset S ⊂ R is allowed to contain zero-divisors. Contents Introduction 1 1. Preliminaries 3 2. S-divisible and S-torsion modules 5 3. S-strongly flat modules 7 4. S-h-local rings 9 5. t-contramodules 14 6. S-h-nil rings 16 7. S-almost perfect rings 22 8. The condition P1 = F1 25 References 28 Introduction Let R be a commutative ring and Q its total ring of quotients. An 1 R-module C is said to be weakly cotorsion if ExtR(Q; C) = 0. An R-module 1 F is strongly flat if ExtR(F; C) = 0 for all weakly cotorsion R-modules C. This definition first appeared in the paper [Trl01]. The problem of char- acterizing commutative domains R for which the class of all strongly flat modules is covering was posed in lecture notes [Trl00]. This problem was solved in the series of papers [BS04, BS02]. It was shown that, for a commutative domain R, the class of all strongly flat R-modules is covering if and only if it coincides with the class of all flat R-modules, and this holds if and only if all the quotient rings of R by nonzero ideals 2010 Mathematics Subject Classification. -
On Types of Elliptic Pseudoprimes
journal of Groups, Complexity, Cryptology Volume 13, Issue 1, 2021, pp. 1:1–1:33 Submitted Jan. 07, 2019 https://gcc.episciences.org/ Published Feb. 09, 2021 ON TYPES OF ELLIPTIC PSEUDOPRIMES LILJANA BABINKOSTOVA, A. HERNANDEZ-ESPIET,´ AND H. Y. KIM Boise State University e-mail address: [email protected] Rutgers University e-mail address: [email protected] University of Wisconsin-Madison e-mail address: [email protected] Abstract. We generalize Silverman's [31] notions of elliptic pseudoprimes and elliptic Carmichael numbers to analogues of Euler-Jacobi and strong pseudoprimes. We inspect the relationships among Euler elliptic Carmichael numbers, strong elliptic Carmichael numbers, products of anomalous primes and elliptic Korselt numbers of Type I, the former two of which we introduce and the latter two of which were introduced by Mazur [21] and Silverman [31] respectively. In particular, we expand upon the work of Babinkostova et al. [3] on the density of certain elliptic Korselt numbers of Type I which are products of anomalous primes, proving a conjecture stated in [3]. 1. Introduction The problem of efficiently distinguishing the prime numbers from the composite numbers has been a fundamental problem for a long time. One of the first primality tests in modern number theory came from Fermat Little Theorem: if p is a prime number and a is an integer not divisible by p, then ap−1 ≡ 1 (mod p). The original notion of a pseudoprime (sometimes called a Fermat pseudoprime) involves counterexamples to the converse of this theorem. A pseudoprime to the base a is a composite number N such aN−1 ≡ 1 mod N. -
On Harada Rings and Serial Artinian Rings
Vietnam Journal of Mathematics 36:2(2008) 229–238 Vietnam Journal of MATHEMATICS © VAST 2008 On Harada Rings and Serial Artinian Rings Thanakarn Soonthornkrachang1, Phan Dan2, Nguyen Van Sanh3, and Kar Ping Shum 4 1,3Department of Mathematics, Mahidol University, Bangkok 10400, Thailand 2Department of Mathematics, University of Transport of Ho Chi Minh City, Vietnam 4Department of Mathematics, The University of Hong Kong, Hong Kong, China (SAR) Received November 18, 2007 Abstract. A ring R is called a right Harada ring if it is right Artinian and every non-small right R-module contains a non-zero injective submodule. The first result in our paper is the following: Let R be a right perfect ring. Then R is a right Harada ring if and only if every cyclic module is a direct sum of an injective module and a small module; if and only if every local module is either injective or small. We also prove that a ring R is QF if and only if every cyclic module is a direct sum of a projective injective module and a small module; if and only if every local module is either projective injective or small. Finally, a right QF-3 right perfect ring R is serial Artinian if and only if every right ideal is a direct sum of a projective module and a singular uniserial module. 2000 Mathematics Subject Classification: 16D50, 16D70, 16D80. Keywords: Harada ring, Artinian Ring, Small module, Co-small module. 1. Introduction and Preliminaries Throughout this paper, all rings are associative rings with identity and all right R-modules are unitary. -
On Distribution of Semiprime Numbers
ISSN 1066-369X, Russian Mathematics (Iz. VUZ), 2014, Vol. 58, No. 8, pp. 43–48. c Allerton Press, Inc., 2014. Original Russian Text c Sh.T. Ishmukhametov, F.F. Sharifullina, 2014, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2014, No. 8, pp. 53–59. On Distribution of Semiprime Numbers Sh. T. Ishmukhametov* and F. F. Sharifullina** Kazan (Volga Region) Federal University, ul. Kremlyovskaya 18, Kazan, 420008 Russia Received January 31, 2013 Abstract—A semiprime is a natural number which is the product of two (possibly equal) prime numbers. Let y be a natural number and g(y) be the probability for a number y to be semiprime. In this paper we derive an asymptotic formula to count g(y) for large y and evaluate its correctness for different y. We also introduce strongly semiprimes, i.e., numbers each of which is a product of two primes of large dimension, and investigate distribution of strongly semiprimes. DOI: 10.3103/S1066369X14080052 Keywords: semiprime integer, strongly semiprime, distribution of semiprimes, factorization of integers, the RSA ciphering method. By smoothness of a natural number n we mean possibility of its representation as a product of a large number of prime factors. A B-smooth number is a number all prime divisors of which are bounded from above by B. The concept of smoothness plays an important role in number theory and cryptography. Possibility of using the concept in cryptography is based on the fact that the procedure of decom- position of an integer into prime divisors (factorization) is a laborious computational process requiring significant calculating resources [1, 2]. -
Cracking RSA with Various Factoring Algorithms Brian Holt
Cracking RSA with Various Factoring Algorithms Brian Holt 1 Abstract For centuries, factoring products of large prime numbers has been recognized as a computationally difficult task by mathematicians. The modern encryption scheme RSA (short for Rivest, Shamir, and Adleman) uses products of large primes for secure communication protocols. In this paper, we analyze and compare four factorization algorithms which use elementary number theory to assess the safety of various RSA moduli. 2 Introduction The origins of prime factorization can be traced back to around 300 B.C. when Euclid's theorem and the unique factorization theorem were proved in his work El- ements [3]. For nearly two millenia, the simple trial division algorithm was used to factor numbers into primes. More efficient means were studied by mathemeticians during the seventeenth and eighteenth centuries. In the 1640s, Pierre de Fermat devised what is known as Fermat's factorization method. Over one century later, Leonhard Euler improved upon Fermat's factorization method for numbers that take specific forms. Around this time, Adrien-Marie Legendre and Carl Gauss also con- tributed crucial ideas used in modern factorization schemes [3]. Prime factorization's potential for secure communication was not recognized until the twentieth century. However, economist William Jevons antipicated the "one-way" or ”difficult-to-reverse" property of products of large primes in 1874. This crucial con- cept is the basis of modern public key cryptography. Decrypting information without access to the private key must be computationally complex to prevent malicious at- tacks. In his book The Principles of Science: A Treatise on Logic and Scientific Method, Jevons challenged the reader to factor a ten digit semiprime (now called Jevon's Number)[4]. -
Which Polynomials Represent Infinitely Many Primes?
Global Journal of Pure and Applied Mathematics. ISSN 0973-1768 Volume 14, Number 1 (2018), pp. 161-180 © Research India Publications http://www.ripublication.com/gjpam.htm Which polynomials represent infinitely many primes? Feng Sui Liu Department of Mathematics, NanChang University, NanChang, China. Abstract In this paper a new arithmetical model has been found by extending both basic operations + and × into finite sets of natural numbers. In this model we invent a recursive algorithm on sets of natural numbers to reformulate Eratosthene’s sieve. By this algorithm we obtain a recursive sequence of sets, which converges to the set of all natural numbers x such that x2 + 1 is prime. The corresponding cardinal sequence is strictly increasing. Test that the cardinal function is continuous with respect to an order topology, we immediately prove that there are infinitely many natural numbers x such that x2 + 1 is prime. Based on the reasoning paradigm above we further prove a general result: if an integer valued polynomial of degree k represents at least k +1 positive primes, then this polynomial represents infinitely many primes. AMS subject classification: Primary 11N32; Secondary 11N35, 11U09,11Y16, 11B37. Keywords: primes in polynomial, twin primes, arithmetical model, recursive sieve method, limit of sequence of sets, Ross-Littwood paradox, parity problem. 1. Introduction Whether an integer valued polynomial represents infinitely many primes or not, this is an intriguing question. Except some obvious exceptions one conjectured that the answer is yes. Example 1.1. Bouniakowsky [2], Schinzel [35], Bateman and Horn [1]. In this paper a recursive algorithm or sieve method adds some exotic structures to the sets of natural numbers, which allow us to answer the question: which polynomials represent infinitely many primes? 2 Feng Sui Liu In 1837, G.L. -
Variations on Euclid's Formula for Perfect Numbers
1 2 Journal of Integer Sequences, Vol. 13 (2010), 3 Article 10.3.1 47 6 23 11 Variations on Euclid’s Formula for Perfect Numbers Farideh Firoozbakht Faculty of Mathematics & Computer Science University of Isfahan Khansar Iran [email protected] Maximilian F. Hasler Laboratoire CEREGMIA Univ. Antilles-Guyane Schoelcher Martinique [email protected] Abstract We study several families of solutions to equations of the form σ(n)= An + B(n), where B is a function that may depend on properties of n. 1 Introduction We recall that perfect numbers (sequence A000396 of Sloane’s Encyclopedia [8]) are defined as solutions to the equation σ(x) = 2 x, where σ(x) denotes the sum of all positive divisors of x, including 1 and x itself. Euclid showed around 300 BCE [2, Proposition IX.36] that q−1 q all numbers of the form x = 2 Mq, where Mq = 2 1 is prime (A000668), are perfect numbers. − While it is still not known whether there exist any odd perfect numbers, Euler [3] proved a converse of Euclid’s proposition, showing that there are no other even perfect numbers (cf. A000043, A006516). (As a side note, this can also be stated by saying that the even perfect 1 numbers are exactly the triangular numbers (A000217(n) = n(n + 1)/2) whose indices are Mersenne primes A000668.) One possible generalization of perfect numbers is the multiply or k–fold perfect numbers (A007691, A007539) such that σ(x) = k x [1, 7, 9, 10]. Here we consider some modified equations, where a second term is added on the right hand side. -
On Rings Whose Flat Modules Form a Grothendieck Category
COLLOQUIUMMATHEMATICUM VOL. 73 1997 NO. 1 ON RINGS WHOSE FLAT MODULES FORM A GROTHENDIECK CATEGORY BY J . L . G A R C I A (MURCIA) AND D. S I M S O N (TORUN)´ 1. Introduction. Throughout this paper, by a ring we shall mean “a ring with enough idempotents” in the sense of [4] and [26, p. 464], that is, an associative ring R containing a set {eλ}λ∈Λ of pairwise orthogonal idempotent elements eλ, λ ∈ Λ, such that M M (1.1) R = Reλ = eλR. λ∈Λ λ∈Λ We say that the ring R is unitary if it has an identity element 1. In this case the set Λ is finite. By a right R-module we shall always mean a right R-module M which is unitary, that is, MR = M. We denote by Mod(R) the category of all unitary right R-modules, and thus Mod(Rop) will stand for the category of left R-modules. The full subcategory of Mod(R) formed by all finitely generated projective modules will be denoted by proj(R). A right R-module M is flat if the tensor product functor op M ⊗R (−) : Mod(R ) → Ab is exact. The full subcategory of Mod(R) consisting of all flat right R- modules will be denoted by Fl(R). For convenience, we introduce the fol- lowing definition. L L Definition 1.2. A ring R = λ∈Λ Reλ = λ∈Λ eλR as in (1.1) is right panoramic if the category Fl(R) of flat right R-modules is abelian, or, equivalently, if Fl(R) is a Grothendieck category.