Subject(s): Mathematics–Number Theory
1. Let R be a ring and let x ∈R. We inductively define1 multiplication of elements of R by
positive integers: for x ∈R, we set 1x = x (here 1 = 1Zis the multiplicative identity of Z),
and for n ∈N, we set (n + 1)x = nx + x.
(a) Prove that (m + n)x = mx + nx for all m,n ∈Nand x ∈R.
(b) Prove that n(x + y) = nx + ny for all n ∈Nand x,y ∈R.
(c) Prove that (mn)x = m(nx) for all m,n ∈Nand x ∈R.
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