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Nilanjan Bhowmick AIR 3, CSIR NET (Earth Science)
- IIT JAM
- Mathematics (MA)
(desoibe all the ubiny ef th vring of intgeet rieckgecs
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Anonymous User Best Answer
am going to admit non-unital subrings of Z to my discussion; if every subring contains 1, then every subring is equal to Z and there is not much more to be written. I claim that if S⊆Z(1) is an additive subgroup, then it is of the form S=sZ,(2), for some s∈Z, for if S≠{0},(3), there is some 0≠s∈S; since SS is a subgroup, s∈S⟺−s∈S,(4), so we may without loss of generality assume that s>0;(5) since SS has positive elements, it has a smallest such; we may assume that ss is the same. Then clearly ns∈S,∀n∈Z;(6); this may be seen by simply adding ss or −s to itself nn times. Thus sZ⊂S;(7) now if there is some t∈S∖sZ,(8), we may let m∈Zbe the largest integer with ms