Vol. 3, Issue 2, Part G (2017)

In this research paper we explained and see that the results for ring of operators on Hilbert space is same as the ring, whenever we suppose H be a Hilbert space and R be a set of operators on H given by R={Ti:Ti2=O, TiTj=O for all i,j} then (R,+) is an additive abelian group and (R,.) is multiplicative group as well as the algebraic structure (R, +,.) forms a ring. Also a special case achieved that If (R,+,.) is a Boolean ring then it is commutative ring. Also we got a beautiful result as (R,+,.) is commutative if and only if (Ti + Tj)2 = Ti2 + 2TiTj + Tj2. We got result on homomorphism and isomorphism also as Let R and R1 be rings of operators on a Hilbert space H and f: R→R1 be a mapping defined by f(Ti)=Ti, then f is Homomorphism as well as Isomorphism. Thus here we have seen some important results of the algebraic structure Ring is the same of the Ring of operators on a Hilbert Space.