Let R be a ring in which OR 1R. For n € Z define nl, as: 1R+1R+...+1R if n> 0 n times n1R := char R := 0 -nlR { if n = 0 Prove: (i) S:= {n¹R|n € Z} is a subring of R; (ii) SZ or S≈ Zm for some m € Z>o; The characteristic of R, denoted char R is defined as if n <0. 0 if SZ m if S Zm. (iii) If F is a field, prove that char F either equals 0 or is a prime. (iv) Express char R' × R" in terms of char R' and char R".

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