. Let R be a ring and let x € R. We inductively define¹ multiplication of elements of R by ositive integers: for x ER, we set 1x = x (here 1 = 1z is the multiplicative identity of Z), nd for ne N, we set (n+1)x= nx+x. (a) Prove that (m+n)x= mx +nx for all m, n € N and x € R. (b) Prove that n(x + y) = nx+ny for all n € N and x, y € R. (c) Prove that (mn)x= m(nx) for all m, n € N and x € R. ote that it is not the case that these properties are special cases of the distributive and ssociative properties in the ring R, because m and n are positive integers, and cannot be ssumed to be elements of R, an arbitrary ring. With the notation of the preceding problem, we have 2r= (1+1)r= lr+r=r+r

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