AARP14

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Prove that the center Z\, of a ring R\, is a subring containing the identity.


The subring criteria for Z\, is Z\neq \emptyset \, and \forall a,b\in Z,a-b\in Z,ab\in Z\,.


1r=r1=r\forall r\in R\, so 1\in Z\, and Z\neq \emptyset \,.


For some a,b\in Z,(a-b)r=(ar)-(br)=(ra)-(rb)=r(a-b)\forall r\in R\, so a-b\in Z\,.


Also (ab)r=a(br)=a(rb)=(ar)b=(ra)b=r(ab)\, so ab\in Z\,.


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