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K. Shimizu - One of the best experts on this subject based on the ideXlab platform.
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Residue arithMetic circuits using a signed-digit nuMber representation
2000 IEEE International Symposium on Circuits and Systems (ISCAS), 2000Co-Authors: K. ShimizuAbstract:A new concept on residue arithMetic using a radix-2 signed-digit (SD) nuMber representation is presented, by which MeMoryless residue arithMetic circuits using SD adders can be iMpleMented. For a given Modulus M, 2/sup p/-1/spl les/M/spl les/2/sup p/+2/sup p-1/-1, in a residue nuMber systeM (RNS), the Modulo M addition is perforMed by using two p-digit SD adders. Thus, the Module M addition tiMe is independent of the word length of operands. When M=2/sup p/ or M=2/sup p//spl plusMn/1, especially, the Module M addition is iMpleMented by only using one SD adder. Moreover, a Module M Multiplier can be constructed using a binary Modulo M SD adder tree, so that the Modulo M Multiplication can be perforMed in a tiMe proportional to log/sub 2/p.
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Residue arithMetic circuits based on signed-digit nuMber representation and the VHDL iMpleMentation
Proceedings Ninth Great Lakes Symposium on VLSI, 1999Co-Authors: K. ShimizuAbstract:Residue arithMetic circuits based on radix-2 signed-digit (SD) nuMber representation, using integers 2/sup p/ and 2/sup p//spl plusMn/1 as Moduli of residue nuMber systeM (RNS), are presented. The Modulo M addition, M=2/sup p/ or M=2/sup p//spl plusMn/1, is perforMed by a carry-free SD adder and the Module M Multiplier is constructed using a binary Modulo M SD adder tree. The iMpleMentation for the residue arithMetic circuits with VHDL description is proposed. The Module M adders and Multipliers have about 530 and 5000 gates, respectively, in cases of M=2/sup 16//spl plusMn/1.
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Residue arithMetic Multiplier based on the radix-4 signed-digit Multiple-valued arithMetic circuits
Proceedings Twelfth International Conference on VLSI Design. (Cat. No.PR00013), 1999Co-Authors: K. ShimizuAbstract:Residue arithMetic Multiplier based on the radix-4 signed-digit arithMetic is presented. Conventional residue arithMetic circuits have been designed using binary nuMber arithMetic systeM, but the carry propagation arises which liMits the speed of arithMetic operations in residue Modules. In this paper, two radix-4 signed-digit (SD) nuMber representations, (-2,-1,0,1,2) and (-3,-2,-1,0,1,2,3), are introduced. The forMer is used for the input and output, and the latter for the inner arithMetic circuit of the presented Multiplier. So that, by using integers 4/sup P/ and 4/sup P//spl plusMn/1 as Moduli of residue nuMber systeM (RNS), where p is a positive integer, both the partial product generating circuit and the circuit for suM of the partial products in the Multiplier can be efficiently constructed based on the SD nuMber representations. The Module M addition, M=4/sup P/ or M=4/sup P//spl plusMn/1, can be perforMed by an SD adder or an end-around-carry SD adder with the Multiple-valued circuits and the addition tiMe is independent of the word length of operands. The Modulo M Multiplier can be coMpactly constructed using a binary Modulo M SD adder tree based on the Multiple-valued addition circuits, and consequently the Module M Multiplication is perforMed in O(log/sub 2/p) tiMe.
Abdullah Harmanci - One of the best experts on this subject based on the ideXlab platform.
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DECOMPOSITIONS OF ModuleS SUPPLEMENTED RELATIVE TO A TORSION THEORY
International Journal of Mathematics, 2020Co-Authors: M. Tamer KoŞan, Abdullah HarmanciAbstract:Let R be a ring, M a right R-Module and a hereditary torsion theory in Mod-R with associated torsion functor τ for the ring R. Then M is called τ-suppleMented when for every subModule N of M there exists a direct suMMand K of M such that K ≤ N and N/K is τ-torsion Module. In [4], M is called alMost τ-torsion if every proper subModule of M is τ-torsion. We present here soMe properties of these classes of Modules and look for answers to the following questions posed by the referee of the paper [4]: (1) Let a Module M = M′ ⊕ M″ be a direct suM of a seMisiMple Module M′ and τ-suppleMented Module M″. Is M τ-suppleMented? (2) Can one find a non-stable hereditary torsion theory τ and τ-suppleMented Modules M′ and M″ such that M′ ⊕ M″ is not τ-suppleMented? (3) Can one find a stable hereditary torsion theory τ and a τ-suppleMented Module M such that M/N is not τ-suppleMented for soMe subModule N of M? (4) Let τ be a non-stable hereditary torsion theory and the Module M be a finite direct suM of alMost τ-torsion subModules. Is M τ-suppleMented? (5) Do you know an exaMple of a torsion theory τ and a τ-suppleMented Module M with τ-torsion subModule τ(M) such that M/τ(M) is not seMisiMple?
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Rickart Modules relative to singular subModule and dual Goldie torsion theory
Journal of Algebra and Its Applications, 2016Co-Authors: Burcu Ungor, Sait Halicioglu, Abdullah HarmanciAbstract:Let R be an arbitrary ring with identity and M a right R-Module with the ring S = EndR(M) of endoMorphisMs of M. The notion of an F-inverse split Module M, where F is a fully invariant subModule of M, is defined and studied by the present authors. This concept produces Rickart subModules of Modules in the sense of Lee, Rizvi and RoMan. In this paper, we consider the subModule F of M as Z(M) and Z∗(M), and investigate soMe properties of Z(M)-inverse split Modules and Z∗(M)-inverse split Modules M. Results are applied to characterize rings R for which every free (projective) right R-Module M is F-inverse split for the preradicals such as Z(⋅) and Z∗(⋅).
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Modules in Which Inverse IMages of SoMe SubModules are Direct SuMMands
Communications in Algebra, 2016Co-Authors: Burcu Ungor, Sait Halicioglu, Abdullah HarmanciAbstract:Let R be an arbitrary ring with identity and M a right R-Module with S = EndR(M). Let F be a fully invariant subModule of M. We call M an F-inverse split Module if f−1(F) is a direct suMMand of M for every f ∈ S. This work is devoted to investigation of various properties and characterizations of an F-inverse split Module M and to show, aMong others, the following results: (1) the Module M is F-inverse split if and only if M = F ⊕ K where K is a Rickart Module; (2) for every free R-Module M, there exists a fully invariant subModule F of M such that M is F-inverse split if and only if for every projective R-Module M, there exists a fully invariant subModule F of M such that M is F-inverse split; and (3) Every R-Module M is Z2(M)-inverse split and Z2(M) is projective if and only if R is seMisiMple.
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On soMe classes of Modules
Czechoslovak Mathematical Journal, 2000Co-Authors: Gonca Güngöroglu, Abdullah HarmanciAbstract:The aiM of this paper is to investigate quasi-corational, coMonoforM, copolyforM and α-(co)atoMic Modules. It is proved that for an ordinal α a right R-Module M is α-atoMic if and only if it is α-coatoMic. And it is also shown that an α-atoMic Module M is quasi-projective if and only if M is quasi-corationally coMplete. SoMe other results are developed.
Ebrahim Hashemi - One of the best experts on this subject based on the ideXlab platform.
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Extensions of Baer and quasi-Baer Modules
Bulletin of The Iranian Mathematical Society, 2011Co-Authors: Ebrahim HashemiAbstract:We study the relationships between the Baer, quasi- Baer and p.q.-Baer property of an R-Module M and the polynoMial extensions of Module M. As a consequence of our results, we obtain soMe results of (C.Y. Hong, N.K. KiM and T.K. Kwak, J. Pure Appl. Algebra 151 (2000) 215-226.) and (E. HasheMi and A. Moussavi, Acta Math. Hungar. 107 (2005) 207-224.).
Ahmad Moussavi - One of the best experts on this subject based on the ideXlab platform.
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PolynoMial extensions of Modules with the quasi-Baer property
Journal of Algebra, 2020Co-Authors: P. Amirzadeh Dana, Ahmad MoussaviAbstract:Abstract In this paper it is shown that, for a Module M over a ring R with S = E n d R ( M ) , the endoMorphisM ring of the R [ x ] -Module M [ x ] is isoMorphic to a subring of S [ [ x ] ] . Also the endoMorphisM ring of the R [ [ x ] ] -Module M [ [ x ] ] is isoMorphic to S [ [ x ] ] . As a consequence, we show that for a Module M R and an arbitrary noneMpty set of not necessarily coMMuting indeterMinates X, M R is quasi-Baer if and only if M [ X ] R [ X ] is quasi-Baer if and only if M [ [ X ] ] R [ [ X ] ] is quasi-Baer if and only if M [ x ] R [ x ] is quasi-Baer if and only if M [ [ x ] ] R [ [ x ] ] is quasi-Baer. Moreover, a Module M R with IFP, is Baer if and only if M [ x ] R [ x ] is Baer if and only if M [ [ x ] ] R [ [ x ] ] is Baer. It is also shown that, when M R is a finitely generated Module, and every seMicentral ideMpotent in S is central, then M [ [ X ] ] R [ [ X ] ] is endo-p.q.-Baer if and only if M [ [ x ] ] R [ [ x ] ] is endo-p.q.-Baer if and only if M R is endo-p.q.-Baer and every countable faMily of fully invariant direct suMMand of M has a generalized countable join. Our results extend several existing results.
Mehdi Sadik - One of the best experts on this subject based on the ideXlab platform.
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Pure Injective Modules Relative to Torsion Theories
Journal of Advances in Mathematics, 2014Co-Authors: Mehdi S. Abbas, Mohanad Farhan Hamid, Mehdi SadikAbstract:Let be a hereditary torsion theory on the category Mod-R of right R-Modules. A right R-Module M is called pure -injective if it is injective with respect to every pure exact sequence having a -torsion cokernel. Every Module has a pure -injective envelope. A Module M is called purely quasi -injective if it is fully invariant in its pure -injective envelope. A Module M is called quasi pure -injective if Maps froM dense pure subModules of M into M are extendable to endoMorphisMs of M. The class of pure -injective Modules is properly contained in the class of purely quasi -injective Modules which is in turn properly contained in the class of quasi pure -injectives. A torsion theoretic version of each of the concepts of regular and pure seMisiMple rings is characterized using the above generalizations of pure injectivity.
