Property 1: ϕ(nr)=nϕ(r)https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003182306/23794b3f-5da1-4165-b755-bb15ac52f817/content/math15_1.tif"/> holds because a ring is a group under addition. To prove that ϕ(rn)=(ϕ(r))nhttps://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003182306/23794b3f-5da1-4165-b755-bb15ac52f817/content/math15_2.tif"/> we note that by induction, ϕ(rn)=ϕ(rn−1r)=ϕ(rn−1)ϕ(r)=ϕ(r)n−1ϕ(r)=ϕ(r)nhttps://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003182306/23794b3f-5da1-4165-b755-bb15ac52f817/content/math15_3.tif"/>.