Let R be a commutative ring.
For each a ∈ R, define πa: R → R by πa(x) = ax. Prove that πa is a group homomorphism from R to R.
Suppose a ≠ 0. Prove that πa is injective ⇔ a is not a divisor of zero.
Suppose R is a ring with unity. Prove that πa is surjective ⇔ a is a unit.
Let A = {πa | a ∈ R}. Define addition and multiplication on A by (πa + πb)(x) = πa(x) + πb(x) and πaπb = πa ◦ πb. Prove that (A, +, ◦) is a ring.
Define ϕ: R → A by ϕ(a) = πa. Prove that ϕ is a ring homomorphism.
If R is a ring with unity, prove that ϕ is an isomorphism. If R has no divisors of zero, prove that ϕ is an isomorphism.