User:Manasapriya/Sandbox

Division Ring A ring (R, +, .) with unity is said to be division ring if the set of all non zero elements of R forms a group w.r.t '.' A division ring need not be commutative if the division ring is commutative then it is said to be a field. Example for division ring which is not a field. Most of the division rings found are commutative. Historically,one of the most non-commutative ring was discovered by sir William Rowell Hamilton(1805-1865). This ring which is a division ring which is continues play an important role in the subsequent of certain areas of mathematics and physics. The (real) Hamilton Quaternions Let H be the collection of elements of the form a+bi+cj+dk where a,b,c,d belongs to R (loosely polynomials in 1 i j k with real coefficient). Here addition is defined component wise and multiplication is defined by expanding distribution law(being careful about the order of terms) and simplifying i²=j²=k²=-1,ij=-ji=k,jk=-kj=i, ki=-ik=j(where the real number coefficients commute with i,j and k). To prove that H is a division ring which is not commutative. First we show that (H,+) is an abelian group.

i) for a' € H      a" € H   a'+a" € H   Closure is satisfied ii) for all a',a",a € H  Then a'+(a"+a)=(a'+a")+a'''  Therefore H is association under addition. iii) additive identity  0=0i+0j+0k € H such that   a'+0=0+a' for all a' € H. iv) Inverse law/additive inverse For a' € H, a' is not equal to H, for -a' € H Such that a'+(-a')=(-a')+a'=0 V) commutativity for all a',a" € H  Then a'+a"=a"+a'   Now we show that (H,.)is a group i) for all a', a" € H such that a'.a" € H  Closure law is satisfied. ii) for all a', a",a € H such that  (a'.a").a=a'.(a".a) Therefore associative law is satisfied. iii) for any a',a",a € H Such that a'.(a"+a)=(a'+a").a Therefore distributivity is satisfied. iv) multiplicative identity/inverse law For 1=1+0i+0j+0k € R such that  a'.1=1.a'=a' is satisfied V) for a' € H,a' is not equal to '0' (a')-¹ € H such that a'. (a')-¹ =1 Inverse law is satisfied. Vi) for a',a" € H such that a'. a" is not equal to a".a' Because in simplification i²=j²=k²=-1,                           ij=-ji=k,                           jk=-kj=i,                           ki=-ik=j,    Commutativity fails    (H,+,.)is a division ring which is not commutative. Hence not a field.