IBN Rings and Orderings on Grothendieck Groups

Let R be a ring with an identity element.R∈IBN means that Rm■Rn implies m=n,R ∈IBN1 means that Rm ■Rn⊕K implies m≥n,and R ∈IBN2 means that Rm■Rm⊕K implies K=0.In this paper we give some characteristic properties of IBN1 and IBN2,with orderings o the Grothendieck groups.In addition,we obtain the following results:（1）If R ∈IBM1 and all finitely generated projective left R-modules are stably free,then the Grothendieck group Ko（R）is a totally ordered abelian group.（2）If the pre-ordering of the Grothendieck group Ko（R）of a ring R is a partial ordering,then R ∈IBM1 or Ko（R）=0.

Frequency equation for the submicron CMOS ring oscillator using the first order characterization

By utilizing the first order behavior of the device,an equation for the frequency of operation of the submicron CMOS ring oscillator is presented.A 5-stage ring oscillator is utilized as the initial design,with different Beta ratios,for the computation of the operating frequency.Later on,the circuit simulation is performed from 5-stage till 23-stage,with the range of oscillating frequency being 3.0817 and 0.6705 GHz respectively.It is noted that the output frequency is inversely proportional to the square of the device length,and when the value of Beta ratio is used as 2.3,a difference of 3.64%is observed on an average,in between the computed and the simulated values of frequency.As an outcome,the derived equation can be utilized,with the inclusion of an empirical constant in general,for arriving at the ring oscillator circuit’s output frequency.