41 Elementary Number Theory 92 41.1 Subgroups of the Integers . . . . . . . . . . . . . . . . . . . . 92 41.2 Greatest Common Divisors . . . . . . . . . . . . . . . . . . . . 92 41.3 The Euclidean Algorithm . . . . . . . . . . . . . . . . . . . . . 93 41.4 Prime Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . 96 41.5 The Fundamental Theorem of Arithmetic . . . . . . . . . . . . 97 4...

Let R be a (nonzero commutative unital) ring. If I and J are ideals of R such that R/I ⊕R/J is a cyclic R-module, then I + J = R. The rings R such that R/I ⊕R/J is a cyclic R-module for all distinct nonzero proper ideals I and J of R are the following three types of principal ideal rings: fields, rings isomorphic to K ×L for the fields K and L, and special principal ideal rings (R,M) such thatM...

In estimating frequencies given that the signal waveforms are undersampled multiple times, Xia and his collaborators proposed to use a generalized version of Chinese remainder Theorem (CRT), where the moduli are dm1, dm2, · · · , dmk with m1,m2, · · · ,mk being pairwise coprime. If the errors of the corrupted remainders are within d 4 , their schemes are able to construct an approximation of th...