For any positive power n of a prime p we find a complete set of generating relations between the elements [r] = rⁿ - r and p·1 of a unitary commutative ring.
2
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We prove that generating relations between the elements [r] = r²-r of a commutative ring are the following: [r+s] = [r]+[s]+rs[2] and [rs] = r²[s]+s[r].
3
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We find complete sets of generating relations between the elements [r] = rⁿ - r for $n = 2^l$ and for n = 3. One of these relations is the n-derivation property [rs] = rⁿ[s] + s[r], r,s ∈ R.