On indecomposable decompositions of CS-modules II

We have built a parametric amplifier with a Josephson field effect transistor (JoFET) as the active element. The device's resonant frequency is field-effect tunable over a range of 2 GHz. The JoFET amplifier has 20 dB of gain, 4 MHz of instantaneous bandwidth, and a 1 dB compression point of -125.5 dBm when operated at a fixed resonance frequency. 

Gate-Tunable Superconductor-Semiconductor Parametric Amplifier

A module M is called a CS module if every submodule of M is essential in a direct summand of M. A ring R is said to be right (countably) s-CS if any direct sum of (countably many) copies of the right R-module R is CS. It is shown that for a right countably s-CS ring R the following are equivalent: (i) R is right s-CS, (ii) R has ACC or DCC on projective principal right ideals, (iii) R has finite right uniform dimension and ACC or DCC holds on projective uniform principal right ideals of R, (iv) R is semiperfect. From results of Oshiro [ 12], [13], under these conditions, R is left artinian and QF-3. As a consequence, a ring R is quasi-Frobenius if it is right countably s-CS, semiperfect and no nonzero projective right ideals are contained in its Jacobson radical.

/pdf/a-right-countably-sigma-cs-ring-with-acc-or-dcc-on-2299conp2w.pdf

A right countably sigma-CS ring with ACC or DCC on projective principal right ideals is left Artinian and QF-3

On Some Classes of Artinian Rings

In the early 1980s, Harada introduced extending and lifting properties for modules and, simultaneously, considered two new classes of artinian rings which contain QF-rings and Nakayama rings. The main purpose of this note is to survey his work and discuss its development and influence on the theory of rings and modules.

Theories of Harada in Artinian Rings and Applications to Classical Artinian Rings

In [3], [4] Harada has studied the following conditions: (*) Every non-small right /?-module contains a non-zero injective submodule. (*)* Every non-cosmall right /2-module contains a non-zero projective direct summand. And he has found two new classes of rings which are characterized by ideal theoretic conditions: one is perfect rings with (*) and the other one is semiperfect rings with (*)*. In [9], Oshiro has studied these rings by using the lifting and extending property of modules, and defined //-rings and co-//-rings related to (*) and (*)*, respectively. A ring R is called a right //-ring if R is right artinian and R satisfies (*). Dually, R is called a right co-//-ring if R satisfies (*)* and the ACC on right annihilator ideals. A right /2-module M is said to be an extending module if for any submodule A of M there is a direct summand A* of M containing A such that AR is essential in Aξ. If this *'extending property\" holds only for uniform submodules of M9 so M is called a module with the extending property for uniform modules. The following theorem is proved by Oshiro in [9, Theorem 3.18].

https://projecteuclid.org/download/pdf_1/euclid.ojm/1200781542

Right perfect rings with the extending property on finitely generated free modules

Superconductor-semiconductor hybrid devices are at the heart of several proposed approaches to quantum information processing, but their basic properties remain to be understood. We embed a two-dimensional Al-InAs hybrid system in a resonant microwave circuit, probing the breakdown of superconductivity due to an applied magnetic field. We find a fingerprint from the two-component nature of the hybrid system, and quantitatively compare with a theory that includes the contribution of intraband p±ip pairing in the InAs, as well as the emergence of Bogoliubov-Fermi surfaces due to magnetic field. Separately resolving the Al and InAs contributions allows us to determine the carrier density and mobility in the InAs. 

/pdf/detecting-induced-mml-math-xmlns-mml-http-www-w3-org-1998-1ujmo0bs.pdf

Detecting Induced 
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mi>p</mml:mi><mml:mo>±</mml:mo><mml:mi>i</mml:mi><mml:mi>p</mml:mi></mml:mrow></mml:math>
 Pairing at the Al-InAs Interface with a Quantum Microwave Circuit

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Right perfect rings with the extending property on finitely generated free modules

Detecting Induced <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mi>p</mml:mi><mml:mo>±</mml:mo><mml:mi>i</mml:mi><mml:mi>p</mml:mi></mml:mrow></mml:math> Pairing at the Al-InAs Interface with a Quantum Microwave Circuit